# 38 Group Libraries

When you start GAP3 it already knows several groups. Currently GAP3 initially knows the following groups:

• some basic groups, such as cyclic groups or symmetric groups (see The Basic Groups Library),

• the primitive permutation groups of degree at most 50 (see The Primitive Groups Library),

• the transitive permutation groups of degree at most 15 (see The Transitive Groups Library),

• the solvable groups of size at most 100 (see The Solvable Groups Library),

• the 2-groups of size at most 256 (see The 2-Groups Library),

• the 3-groups of size at most 729 (see The 3-Groups Library),

• the irreducible solvable subgroups of GL(n,p) for n > 1 and pn < 256 (see The Irreducible Solvable Linear Groups Library),

• the finite perfect groups of size at most 106 (excluding 11 sizes) (see The Library of Finite Perfect Groups),

• the irreducible maximal finite integral matrix groups of dimension at most 24 (see Irreducible Maximal Finite Integral Matrix Groups),

• the crystallographic groups of dimension at most 4 (see The Crystallographic Groups Library).

• the groups of order at most 1000 except for 512 and 768 (see The Small Groups Library).

Each of the set of groups above is called a library. The whole set of groups that GAP3 knows initially is called the GAP3 collection of group libraries. There is usually no relation between the groups in the different libraries.

Several of the libraries are accessed in a uniform manner. For each of these libraries there is a so called selection function that allows you to select the list of groups that satisfy given criterias from a library. The example function allows you to select one group that satisfies given criteria from the library. The low-level extraction function allows you to extract a single group from a library, using a simple indexing scheme. These functions are described in the sections Selection Functions, Example Functions, and Extraction Functions.

Note that a system administrator may choose to install all, or only a few, or even none of the libraries. So some of the libraries mentioned below may not be available on your installation.

## 38.1 The Basic Groups Library

CyclicGroup( n )
CyclicGroup( D, n )

In the first form CyclicGroup returns the cyclic group of size n as a permutation group. In the second form D must be a domain of group elements, e.g., Permutations or AgWords, and CyclicGroup returns the cyclic group of size n as a group of elements of that type.

    gap> c12 := CyclicGroup( 12 );
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12) )
gap> c105 := CyclicGroup( AgWords, 5*3*7 );
Group( c105_1, c105_2, c105_3 )
gap> Order(c105,c105.1); Order(c105,c105.2); Order(c105,c105.3);
105
35
7 

AbelianGroup( sizes )
AbelianGroup( D, sizes )

In the first form AbelianGroup returns the abelian group Csizes * Csizes * ... * Csizes[n], where sizes must be a list of positive integers, as a permutation group. In the second form D must be a domain of group elements, e.g., Permutations or AgWords, and AbelianGroup returns the abelian group as a group of elements of this type.

    gap> g := AbelianGroup( AgWords, [ 2, 3, 7 ] );
Group( a, b, c )
gap> Size( g );
42
gap> IsAbelian( g );
true 

The default function GroupElementsOps.AbelianGroup uses the functions CyclicGroup and DirectProduct (see DirectProduct) to construct the abelian group.

ElementaryAbelianGroup( n )
ElementaryAbelianGroup( D, n )

In the first form ElementaryAbelianGroup returns the elementary abelian group of size n as a permutation group. n must be a positive prime power of course. In the second form D must be a domain of group elements, e.g., Permutations or AgWords, and ElementaryAbelianGroup returns the elementary abelian group as a group of elements of this type.

    gap> ElementaryAbelianGroup( 16 );
Group( (1,2), (3,4), (5,6), (7,8) )
gap> ElementaryAbelianGroup( AgWords, 3 ^ 10 );
Group( m59049_1, m59049_2, m59049_3, m59049_4, m59049_5, m59049_6,
m59049_7, m59049_8, m59049_9, m59049_10 ) 

The default function GroupElementsOps.ElementaryAbelianGroup uses CyclicGroup and DirectProduct (see DirectProduct to construct the elementary abelian group.

DihedralGroup( n )
DihedralGroup( D, n )

In the first form DihedralGroup returns the dihedral group of size n as a permutation group. n must be a positive even integer. In the second form D must be a domain of group elements, e.g., Permutations or AgWords, and DihedralGroup returns the dihedral group as a group of elements of this type.

    gap> DihedralGroup( 12 );
Group( (1,2,3,4,5,6), (2,6)(3,5) ) 

PolyhedralGroup( p, q )
PolyhedralGroup( D, p, q )

In the first form PolyhedralGroup returns the polyhedral group of size p * q as a permutation group. p and q must be positive integers and there must exist a nontrivial p-th root of unity modulo every prime factor of q. In the second form D must be a domain of group elements, e.g., Permutations or Words, and PolyhedralGroup returns the polyhedral group as a group of elements of this type.

    gap> PolyhedralGroup( 3, 13 );
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13), ( 2, 4,10)( 3, 7, 6)
( 5,13,11)( 8, 9,12) )
gap> Size( last );
39 

SymmetricGroup( d )
SymmetricGroup( D, d )

In the first form SymmetricGroup returns the symmetric group of degree d as a permutation group. d must be a positive integer. In the second form D must be a domain of group elements, e.g., Permutations or Words, and SymmetricGroup returns the symmetric group as a group of elements of this type.

    gap> SymmetricGroup( 8 );
Group( (1,8), (2,8), (3,8), (4,8), (5,8), (6,8), (7,8) )
gap> Size( last );
40320 

AlternatingGroup( d )
AlternatingGroup( D, d )

In the first form AlternatingGroup returns the alternating group of degree d as a permutation group. d must be a positive integer. In the second form D must be a domain of group elements, e.g., Permutations or Words, and AlternatingGroup returns the alternating group as a group of elements of this type.

    gap> AlternatingGroup( 8 );
Group( (1,2,8), (2,3,8), (3,4,8), (4,5,8), (5,6,8), (6,7,8) )
gap> Size( last );
20160 

GeneralLinearGroup( n, q )
GeneralLinearGroup( D, n, q )

In the first form GeneralLinearGroup returns the general linear group GL( n, q ) as a matrix group. In the second form D must be a domain of group elements, e.g., Permutations or AgWords, and GeneralLinearGroup returns GL( n, q ) as a group of elements of that type.

    gap> g := GeneralLinearGroup( 2, 4 ); Size( g );
GL(2,4)
180 

SpecialLinearGroup( n, q )
SpecialLinearGroup( D, n, q )

In the first form SpecialLinearGroup returns the special linear group SL( n, q ) as a matrix group. In the second form D must be a domain of group elements, e.g., Permutations or AgWords, and SpecialLinearGroup returns SL( n, q ) as a group of elements of that type.

    gap> g := SpecialLinearGroup( 3, 4 ); Size( g );
SL(3,4)
60480 

SymplecticGroup( n, q )
SymplecticGroup( D, n, q )

In the first form SymplecticGroup returns the symplectic group SP( n, q ) as a matrix group. In the second form D must be a domain of group elements, e.g., Permutations or AgWords, and SymplecticGroup returns SP( n, q ) as a group of elements of that type.

    gap> g := SymplecticGroup( 4, 2 ); Size( g );
SP(4,2)
720 

GeneralUnitaryGroup( n, q )
GeneralUnitaryGroup( D, n, q )

In the first form GeneralUnitaryGroup returns the general unitary group GU( n, q ) as a matrix group. In the second form D must be a domain of group elements, e.g., Permutations or AgWords, and GeneralUnitaryGroup returns GU( n, q ) as a group of elements of that type.

    gap> g := GeneralUnitaryGroup( 3, 3 ); Size( g );
GU(3,3)
24192 

SpecialUnitaryGroup( n, q )
SpecialUnitaryGroup( D, n, q )

In the first form SpecialUnitaryGroup returns the special unitary group SU( n, q ) as a matrix group. In the second form D must be a domain of group elements, e.g., Permutations or AgWords, and SpecialUnitaryGroup returns SU( n, q ) as a group of elements of that type.

    gap> g := SpecialUnitaryGroup( 3, 3 ); Size( g );
SU(3,3)
6048 

MathieuGroup( d )

MathieuGroup returns the Mathieu group of degree d as a permutation group. d is expected to be 11, 12, 22, 23, or 24.

    gap> g := MathieuGroup( 12 ); Size( g );
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11), ( 3, 7,11, 8)
( 4,10, 5, 6), ( 1,12)( 2,11)( 3, 6)( 4, 8)( 5, 9)( 7,10) )
95040

## 38.2 Selection Functions

AllLibraryGroups( fun1, val1, fun2, val2, ... )

For each group library there is a selection function. This function allows you to select all groups from the library that have a given set of properties.

The name of the selection functions always begins with All and always ends with Groups. Inbetween is a name that hints at the nature of the group library. For example, the selection function for the library of all primitive groups of degree at most 50 (see The Primitive Groups Library) is called AllPrimitiveGroups, and the selection function for the library of all 2-groups of size at most 256 (see The 2-Groups Library) is called AllTwoGroups.

These functions take an arbitrary number of pairs of arguments. The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group included in the selection, or a list of such values.

For example

    AllPrimitiveGroups( DegreeOperation,  [10..15],
Size,             [1..100],
IsAbelian,        false    );

should return a list of all primitive groups with degree between 10 and 15 and size less than 100 that are not abelian.

Thus the AllPrimitiveGroups behaves as if it was implemented by a function similar to the one defined below, where PrimitiveGroupsList is a list of all primitive groups. Note, in the definition below we assume for simplicity that AllPrimitiveGroups accepts exactly 4 arguments. It is of course obvious how to change this definition so that the function would accept a variable number of arguments.

    AllPrimitiveGroups := function ( fun1, val1, fun2, val2 )
local    groups, g, i;
groups := [];
for i  in [ 1 .. Length( PrimitiveGroupsList ) ] do
g := PrimitiveGroupsList[i];
if      fun1(g) = val1  or IsList(val1) and fun1(g) in val1
and fun2(g) = val2  or IsList(val2) and fun2(g) in val2
then
fi;
od;
return groups;
end; 

Note that the real selection functions are considerably more difficult, to improve the efficiency. Most important, each recognizes a certain set of functions and handles those properties using an index (see About Group Libraries).

## 38.3 Example Functions

OneLibraryGroup( fun1, val1, fun2, val2, ... )

For each group library there is a example function. This function allows you to find one group from the library that has a given set of properties.

The name of the example functions always begins with One and always ends with Group. Inbetween is a name that hints at the nature of the group library. For example, the example function for the library of all primitive groups of degree at most 50 (see The Primitive Groups Library) is called OnePrimitiveGroup, and the example function for the library of all 2-groups of size at most 256 (see The 2-Groups Library) is called OneTwoGroup.

These functions take an arbitrary number of pairs of arguments. The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group returned by the example function, or a list of such values.

For example

    OnePrimitiveGroup( DegreeOperation,  [10..15],
Size,             [1..100],
IsAbelian,        false    );

should return one primitive group with degree between 10 and 15 and size size less than 100 that is not abelian.

Thus the OnePrimitiveGroup behaves as if it was implemented by a function similar to the one defined below, where PrimitiveGroupsList is a list of all primitive groups. Note, in the definition below we assume for simplicity that OnePrimitiveGroup accepts exactly 4 arguments. It is of course obvious how to change this definition so that the function would accept a variable number of arguments.

    OnePrimitiveGroup := function ( fun1, val1, fun2, val2 )
local    g, i;
for i  in [ 1 .. Length( PrimitiveGroupsList ) ] do
g := PrimitiveGroupsList[i];
if      fun1(g) = val1  or IsList(val1) and fun1(g) in val1
and fun2(g) = val2  or IsList(val2) and fun2(g) in val2
then
return g;
fi;
od;
return false;
end; 

Note that the real example functions are considerably more difficult, to improve the efficiency. Most important, each recognizes a certain set of functions and handles those properties using an index (see About Group Libraries).

## 38.4 Extraction Functions

For each group library there is an extraction function. This function allows you to extract single groups from the library.

The name of the extraction function always ends with Group and begins with a name that hints at the nature of the library. For example the extraction function for the library of primitive groups (see The Primitive Groups Library) is called PrimitiveGroup, and the extraction function for the library of all 2-groups of size at most 256 (see The 2-Groups Library) is called TwoGroup.

What arguments the extraction function accepts, and how they are interpreted is described in the sections that describe the individual group libraries.

For example

 PrimitiveGroup( 10, 4 );

returns the 4-th primitive group of degree 10.

The reason for the extraction function is as follows. A group library is usually not stored as a list of groups. Instead a more compact representation for the groups is used. For example the groups in the library of 2-groups are represented by 4 integers. The extraction function hides this representation from you, and allows you to access the group library as if it was a table of groups (two dimensional in the above example).

## 38.5 The Primitive Groups Library

This group library contains all primitive permutation groups of degree at most 50. There are a total of 406 such groups. Actually to be a little bit more precise, there are 406 inequivalent primitive operations on at most 50 points. Quite a few of the 406 groups are isomorphic.

AllPrimitiveGroups( fun1, val1, fun2, val2, ... )

AllPrimitiveGroups returns a list containing all primitive groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function, which will be applied to all groups in the library, and the second being a value or a list of values, that this function must return in order to have this group included in the list returned by AllPrimitiveGroups.

The first argument must be DegreeOperation and the second argument either a degree or a list of degrees, otherwise AllPrimitiveGroups will print a warning to the effect that the library contains only groups with degrees between 1 and 50.

    gap> l := AllPrimitiveGroups( Size, 120, IsSimple, false );
#W  AllPrimitiveGroups: degree automatically restricted to [1..50]
[ S(5), PGL(2,5), S(5) ]
gap> List( l, g -> g.generators );
[ [ (1,2,3,4,5), (1,2) ], [ (1,2,3,4,5), (2,3,5,4), (1,6)(3,4) ],
[ ( 1, 8)( 2, 5, 6, 3)( 4, 9, 7,10), ( 1, 5, 7)( 2, 9, 4)( 3, 8,10)
] ] 

OnePrimitiveGroup( fun1, val1, fun2, val2, ... )

OnePrimitiveGroup returns one primitive group that has the properties given as argument. Each property is specified by passing a pair of arguments, the first being a function, which will be applied to all groups in the library, and the second being a value or a list of values, that this function must return in order to have this group returned by OnePrimitiveGroup. If no such group exists, false is returned.

The first argument must be DegreeOperation and the second argument either a degree or a list of degrees, otherwise OnePrimitiveGroup will print a warning to the effect that the library contains only groups with degrees between 1 and 50.

    gap> g := OnePrimitiveGroup( DegreeOperation,5, IsSolvable,false );
A(5)
gap> Size( g );
60 

AllPrimitiveGroups and OnePrimitiveGroup recognize the following functions and handle them usually quite efficient. DegreeOperation, Size, Transitivity, and IsSimple. You should pass those functions first, e.g., it is more efficient to say AllPrimitiveGroups( Size,120 , IsAbelian,false ) than to say AllPrimitiveGroups( IsAbelian,false, Size,120 ) (see About Group Libraries).

PrimitiveGroup( deg, nr )

PrimitiveGroup returns the nr-th primitive group of degree deg. Both deg and nr must be positive integers. The primitive groups of equal degree are sorted with respect to their size, so for example PrimitiveGroup( deg, 1 ) is the smallest primitive group of degree deg, e.g, the cyclic group of size deg, if deg is a prime. Primitive groups of equal degree and size are in no particular order.

    gap>  g := PrimitiveGroup( 8, 1 );
AGL(1,8)
gap> g.generators;
[ (1,2,3,4,5,6,7), (1,8)(2,4)(3,7)(5,6) ] 

Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.

transitivity:

degree of transitivity of G.

isSharpTransitive:

true if G is sharply G.transitivity-fold transitive and false otherwise.

isKPrimitive:

true if G is k-fold primitive, and false otherwise.

isOdd:

false if G is a subgroup of the alternating group of degree G.degree and true otherwise.

isFrobeniusGroup:

true if G is a Frobenius group and false otherwise.

This library was computed by Charles Sims. The list of primitive permutation groups of degree at most 20 was published in Sim70. The library was brought into GAP3 format by Martin Schönert. He assumes the responsibility for all mistakes.

## 38.6 The Transitive Groups Library

The transitive groups library contains representatives for all transitive permutation groups of degree at most 22. Two permutations groups of the same degree are considered to be equivalent, if there is a renumbering of points, which maps one group into the other one. In other words, if they lie in the save conjugacy class under operation of the full symmetric group by conjugation.

There are a total of 4945 such groups up to degree 22.

AllTransitiveGroups( fun1, val1, fun2, val2, ... )

AllTransitiveGroups returns a list containing all transitive groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function, and the second being a value or a list of values. AllTransitiveGroups will return all groups from the transitive groups library, for which all specified functions have the specified values.

If the degree is not restricted to 22 at most, AllTransitiveGroups will print a warning.

OneTransitiveGroup( fun1, val1, fun2, val2, ... )

OneTransitiveGroup returns one transitive group that has the properties given as argument. Each property is specified by passing a pair of arguments, the first being a function, and the second being a value or a list of values. OneTransitiveGroup will return one groups from the transitive groups library, for which all specified functions have the specified values. If no such group exists, false is returned.

If the degree is not restricted to 22 at most, OneTransitiveGroup will print a warning.

AllTransitiveGroups and OneTransitiveGroup recognize the following functions and get the corresponding properties from a precomputed list to speed up processing:
DegreeOperation, Size, Transitivity, and IsPrimitive. You do not need to pass those functions first, as the selection function picks the these properties first.

TransitiveGroup( deg, nr )

TransitiveGroup returns the nr-th transitive group of degree deg. Both deg and nr must be positive integers. The transitive groups of equal degree are sorted with respect to their size, so for example TransitiveGroup( deg, 1 ) is the smallest transitive group of degree deg, e.g, the cyclic group of size deg, if deg is a prime. The ordering of the groups corresponds to the list in Butler/McKay BM83.

This library was computed by Gregory Butler, John McKay, Gordon Royle and Alexander Hulpke. The list of transitive groups up to degree 11 was published in BM83, the list of degree 12 was published in Roy87, degree 14 and 15 were published in But93.

The library was brought into GAP3 format by Alexander Hulpke, who is responsible for all mistakes.

    gap> TransitiveGroup(10,22);
S(5)[x]2
gap> l:=AllTransitiveGroups(DegreeOperation,12,Size,1440,
> IsSolvable,false);
[ S(6)[x]2, M_10.2(12) = A_6.E_4(12) = [S_6[1/720]{M_10}S_6]2 ]
gap> List(l,IsSolvable);
[ false, false ] 

TransitiveIdentification( G )

Let G be a permutation group, acting transitively on a set of up to 22 points. Then TransitiveIdentification will return the position of this group in the transitive groups library. This means, if G operates on m points and TransitiveIdentification returns n, then G is permutation isomorphic to the group TransitiveGroup(m,n).

    gap> TransitiveIdentification(Group((1,2),(1,2,3)));
2 

## 38.7 The Solvable Groups Library

GAP3 has a library of the 1045 solvable groups of size between 2 and 100. The groups are from lists computed by M. Hall and J. K. Senior (size 64, see HS64), R. Laue (size 96, see Lau82) and J. Neubüser (other sizes, see Neu67).

AllSolvableGroups( fun1, val1, fun2, val2, ... )

AllSolvableGroups returns a list containing all solvable groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function, which will be applied to all the groups in the library, and the second being a value or a list of values, that this function must return in order to have this group included in the list returned by AllSolvableGroups.

    gap> AllSolvableGroups(Size,24,IsNontrivialDirectProduct,false);
[ 12.2, grp_24_11, D24, Q8+S3, Sl(2,3), S4 ] 

OneSolvableGroup( fun1, val1, fun2, val2, ... )

OneSolvableGroup returns a solvable group with the specified properties. Each property is specified by passing a pair of arguments, the first being a function, which will be applied to all the groups in the library, and the second being a value or a list of values, that this function must return in order to have this group returned by OneSolvableGroup. If no such group exists, false is returned.

    gap> OneSolvableGroup(Size,100,x->Size(DerivedSubgroup(x)),10);
false
gap> OneSolvableGroup(Size,24,IsNilpotent,false);
S3x2^2 

AllSolvableGroups and OneSolvableGroup recognize the following functions and handle them usually very efficiently: Size, IsAbelian, IsNilpotent, and IsNonTrivialDirectProduct.

SolvableGroup( size, nr )

SolvableGroup returns the nr-th group of size size in the library. SolvableGroup will signal an error if size is not between 2 and 100, or if nr is larger than the number of solvable groups of size size. The group is returned as finite polycyclic group (see Finite Polycyclic Groups).

    gap> SolvableGroup( 32 , 15 );
Q8x4 

## 38.8 The 2-Groups Library

The library of 2-groups contains all the 2-groups of size dividing 256. There are a total of 58760 such groups, 1 of size 2, 2 of size 4, 5 of size 8, 14 of size 16, 51 of size 32, 267 of size 64, 2328 of size 128, and 56092 of size 256.

AllTwoGroups( fun1, val1, fun2, val2, ... )

AllTwoGroups returns the list of all the 2-groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first is a function that can be applied to each group, the second is either a single value or a list of values that the function must return in order to select that group.

    gap> l := AllTwoGroups( Size, 256, Rank, 3, pClass, 2 );
[ Group( a1, a2, a3, a4, a5, a6, a7, a8 ),
Group( a1, a2, a3, a4, a5, a6, a7, a8 ),
Group( a1, a2, a3, a4, a5, a6, a7, a8 ),
Group( a1, a2, a3, a4, a5, a6, a7, a8 ) ]
gap> List( l, g -> Length( ConjugacyClasses( g ) ) );
[ 112, 88, 88, 88 ] 

OneTwoGroup( fun1, val1, fun2, val2, ... )

OneTwoGroup returns a single 2-group that has the properties given as arguments. Each property is specified by passing a pair of arguments, the first is a function that can be applied to each group, the second is either a single value or a list of values that the function must return in order to select that group.

    gap> g := OneTwoGroup( Size, [64..128], Rank, [2..3], pClass, 5 );
#I  size restricted to [ 64, 128 ]
Group( a1, a2, a3, a4, a5, a6 )
gap> Size( g );
64
gap> Rank( g );
2 

AllTwoGroups and OneTwoGroup recognize the following functions and handle them usually very efficiently. Size, Rank for the rank of the Frattini quotient of the group, and pClass for the exponent-p class of the group. Note that Rank and pClass are dummy functions that can be used only in this context, i.e., they can not be applied to arbitrary groups.

TwoGroup( size, nr )

TwoGroup returns the nr-th group of size size. The group is returned as a finite polycyclic group (see Finite Polycyclic Groups). TwoGroup will signal an error if size is not a power of 2 between 2 and 256, or nr is larger than the number of groups of size size.

Within each size the following criteria have been used, in turn, to determine the index position of a group in the list

1:
increasing generator number;

2:
increasing exponent-2 class;

3:
the position of its parent in the list of groups of appropriate size;

4:
the list in which the Newman and O'Brien implementation of the p-group generation algorithm outputs the immediate descendants of a group.

    gap> g := TwoGroup( 32, 45 );
Group( a1, a2, a3, a4, a5 )
gap> Rank( g );
4
gap> pClass( g );
2
gap> g.abstractRelators;
[ a1^2*a5^-1, a2^2, a2^-1*a1^-1*a2*a1, a3^2, a3^-1*a1^-1*a3*a1,
a3^-1*a2^-1*a3*a2, a4^2, a4^-1*a1^-1*a4*a1, a4^-1*a2^-1*a4*a2,
a4^-1*a3^-1*a4*a3, a5^2, a5^-1*a1^-1*a5*a1, a5^-1*a2^-1*a5*a2,
a5^-1*a3^-1*a5*a3, a5^-1*a4^-1*a5*a4 ] 

Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.

rank:

rank of Frattini quotient of G.

pclass:

exponent-p class of G.

abstractGenerators:

a list of abstract generators of G (see AbstractGenerator).

abstractRelators:

a list of relators of G stored as words in the abstract generators.

Descriptions of the algorithms used in constructing the library data may be found in OBr90,OBr91. Using these techniques, a library was first prepared in 1987 by M.F. Newman and E.A. O'Brien; a partial description may be found in NO89.

The library was brought into the GAP3 format by Werner Nickel, Alice Niemeyer, and E.A. O'Brien.

## 38.9 The 3-Groups Library

The library of 3-groups contains all the 3-groups of size dividing 729. There are a total of 594 such groups, 1 of size 3, 2 of size 9, 5 of size 27, 15 of size 81, 67 of size 243, and 504 of size 729.

AllThreeGroups( fun1, val1, fun2, val2, ... )

AllThreeGroups returns the list of all the 3-groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first is a function that can be applied to each group, the second is either a single value or a list of values that the function must return in order to select that group.

    gap> l := AllThreeGroups( Size, 243, Rank, [2..4], pClass, 3 );;
gap> Length ( l );
33
gap>  List( l, g -> Length( ConjugacyClasses( g ) ) );
[ 35, 35, 35, 35, 35, 35, 35, 243, 99, 99, 51, 51, 51, 51, 51, 51,
51, 51, 99, 35, 243, 99, 99, 51, 51, 51, 51, 51, 35, 35, 35, 35, 35
] 

OneThreeGroup( fun1, val1, fun2, val2, ... )

OneThreeGroup returns a single 3-group that has the properties given as arguments. Each property is specified by passing a pair of arguments, the first is a function that can be applied to each group, the second is either a single value or a list of values that the function must return in order to select that group.

    gap> g := OneThreeGroup( Size, 729, Rank, 4, pClass, [3..5] );
Group( a1, a2, a3, a4, a5, a6 )
gap> IsAbelian( g );
true 

AllThreeGroups and OneThreeGroup recognize the following functions and handle them usually very efficiently. Size, Rank for the rank of the Frattini quotient of the group, and pClass for the exponent-p class of the group. Note that Rank and pClass are dummy functions that can be used only in this context, i.e., they cannot be applied to arbitrary groups.

ThreeGroup( size, nr )

ThreeGroup returns the nr-th group of size size. The group is returned as a finite polycyclic group (see Finite Polycyclic Groups). ThreeGroup will signal an error if size is not a power of 3 between 3 and 729, or nr is larger than the number of groups of size size.

Within each size the following criteria have been used, in turn, to determine the index position of a group in the list

1:
increasing generator number;

2:
increasing exponent-3 class;

3:
the position of its parent in the list of groups of appropriate size;

4:
the list in which the Newman and O'Brien implementation of the p-group generation algorithm outputs the immediate descendants of a group.

    gap> g := ThreeGroup( 243, 56 );
Group( a1, a2, a3, a4, a5 )
gap> pClass( g );
3
gap> g.abstractRelators;
[ a1^3, a2^3, a2^-1*a1^-1*a2*a1*a4^-1, a3^3, a3^-1*a1^-1*a3*a1,
a3^-1*a2^-1*a3*a2*a5^-1, a4^3, a4^-1*a1^-1*a4*a1*a5^-1,
a4^-1*a2^-1*a4*a2, a4^-1*a3^-1*a4*a3, a5^3, a5^-1*a1^-1*a5*a1,
a5^-1*a2^-1*a5*a2, a5^-1*a3^-1*a5*a3, a5^-1*a4^-1*a5*a4 ] 

Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.

rank:

rank of Frattini quotient of G.

pclass:

exponent-p class of G.

abstractGenerators:

a list of abstract generators of G (see AbstractGenerator).

abstractRelators:

a list of relators of G stored as words in the abstract generators.

Descriptions of the algorithms used in constructing the library data may be found in OBr90,OBr91.

The library was generated and brought into GAP3 format by E.A. O'Brien and Colin Rhodes. David Baldwin, M.F. Newman, and Maris Ozols have contributed in various ways to this project and to correctly determining these groups. The library design is modelled on and borrows extensively from the 2-groups library, which was brought into GAP3 format by Werner Nickel, Alice Niemeyer, and E.A. O'Brien.

## 38.10 The Irreducible Solvable Linear Groups Library

The IrredSol group library provides access to the irreducible solvable subgroups of GL(n,p), where n > 1, p is prime and pn < 256. The library contains exactly one member from each of the 370 conjugacy classes of such subgroups.

By well known theory, this library also doubles as a library of primitive solvable permutation groups of non-prime degree less than 256. To access the data in this form, you must first build the matrix group(s) of interest and then call the function
PrimitivePermGroupIrreducibleMatGroup( matgrp )
This function returns a permutation group isomorphic to the semidirect product of an irreducible matrix group (over a finite field) and its underlying vector space.

AllIrreducibleSolvableGroups( fun1, val1, fun2, val2, ... )

AllIrreducibleSolvableGroups returns a list containing all irreducible solvable linear groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function which will be applied to all groups in the library, and the second being a value or a list of values that this function must return in order to have this group included in the list returned by AllIrreducibleSolvableGroups.

    gap> AllIrreducibleSolvableGroups( Dimension, 2,
>                                   CharFFE, 3,
>                                   Size, 8 );
[ Group( [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ] ],
[ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ],
[ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ),
Group( [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ],
[ [ Z(3)^0, Z(3) ], [ Z(3), Z(3) ] ] ),
Group( [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, Z(3) ] ] ) ] 

OneIrreducibleSolvableGroup( fun1, val1, fun2, val2, ... )

OneIrreducibleSolvableGroup returns one irreducible solvable linear group that has the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function which will be applied to all groups in the library, and the second being a value or a list of values that this function must return in order to have this group returned by OneIrreducibleSolvableGroup. If no such group exists, false is returned.

    gap> OneIrreducibleSolvableGroup( Dimension, 4,
>                                  IsLinearlyPrimitive, false );
Group( [ [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ] ],
[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ],
[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ] ] ) 

AllIrreducibleSolvableGroups and OneIrreducibleSolvableGroup recognize the following functions and handle them very efficiently (because the information is stored with the groups and so no computations are needed): Dimension for the linear degree, CharFFE for the field characteristic, Size, IsLinearlyPrimitive, and MinimalBlockDimension. Note that the last two are dummy functions that can be used only in this context. Their meaning is explained at the end of this section.

IrreducibleSolvableGroup( n, p, i )

IrreducibleSolvableGroup returns the i-th irreducible solvable subgroup of GL( n, p ). The irreducible solvable subgroups of

GL(n,p) are ordered with respect to the following criteria:

• increasing size;

• increasing guardian number. If two groups have the same size and guardian, they are in no particular order. (See the library documentation or [Sho92] for the meaning of guardian.)

    gap> g := IrreducibleSolvableGroup( 3, 5, 12 );
Group( [ [ 0*Z(5), Z(5)^2, 0*Z(5) ], [ Z(5)^2, 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5)^2 ] ],
[ [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^0 ],
[ Z(5)^0, 0*Z(5), 0*Z(5) ] ],
[ [ Z(5)^2, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5)^2 ] ],
[ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^2, 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5)^2 ] ],
[ [ Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5) ] ] ) 

Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.

size:

size of G.

isLinearlyPrimitive:

false if G preserves a direct sum decomposition of the underlying vector space, and true otherwise.

minimalBlockDimension:

not bound if G is linearly primitive; otherwise equals the dimension of the blocks in an unrefinable system of imprimitivity for G.

This library was computed and brought into GAP3 format by Mark Short. Descriptions of the algorithms used in computing the library data can be found in Sho92.

## 38.11 The Library of Finite Perfect Groups

The GAP3 library of finite perfect groups provides, up to isomorphism, a list of all perfect groups whose sizes are less than 106 excluding the following:

• For n = 61440, 122880, 172032, 245760, 344064, 491520, 688128, or 983040, the perfect groups of size n have not completely been determined yet. The library neither provides the number of these groups nor the groups themselves. \vspace-1mm

• For n = 86016, 368640, or 737280, the library does not yet contain the perfect groups of size n, it only provides their their numbers which are 52, 46, or 54, respectively. \vspace-2mm

Except for these eleven sizes, the list of altogether 1096 perfect groups in the library is complete. It relies on results of Derek F. Holt and Wilhelm Plesken which are published in their book Perfect Groups HP89. Moreover, they have supplied to us files with presentations of 488 of the groups. In terms of these, the remaining 607 nontrivial groups in the library can be described as 276 direct products, 107 central products, and 224 subdirect products. They are computed automatically by suitable GAP3 functions whenever they are needed.

We are grateful to Derek Holt and Wilhelm Plesken for making their groups available to the GAP3 community by contributing their files. It should be noted that their book contains a lot of further information for many of the library groups. So we would like to recommend it to any GAP3 user who is interested in the groups.

The library has been brought into GAP3 format by Volkmar Felsch.

Like most of the other GAP3 libraries, the library of finite perfect groups provides an extraction function, PerfectGroup. It returns the specified group in form of a finitely presented group which, in its group record, bears some additional information that allows you to easily construct an isomorphic permutation group of some appropriate degree by just calling the PermGroup function.

Further, there is a function NumberPerfectGroups which returns the number of perfect groups of a given size.

The library does not provide a selection or an example function. There is, however, a function DisplayInformationPerfectGroups which allows the display of some information about arbitrary library groups without actually loading the large files with their presentations, and without constructing the groups themselves.

Moereover, there are two functions which allow you to formulate loops over selected library groups. The first one is the NumberPerfectLibraryGroups function which, for any given size, returns the number of groups in the library which are of that size.

The second one is the SizeNumbersPerfectGroups function. It allows you to ask for all library groups which contain certain composition factors. The answer is provided in form of a list of pairs [size,n] where each such pair characterizes the n th library group of size size. We will call such a pair [size,n] the size number of the respective perfect group. As the size numbers are accepted as input arguments by the PerfectGroup and the DisplayInformationPerfectGroups function, you may use their list to formulate a loop over the associated groups.

Now we shall give an individual description of each library function.

NumberPerfectGroups( size )

NumberPerfectGroups returns the number of non-isomorphic perfect groups of size size for each positive integer size up to 106 except for the eight sizes listed at the beginning of this section for which the number is not yet known. For these values as well as for any argument out of range it returns the value -1.

NumberPerfectLibraryGroups( size )

NumberPerfectLibraryGroups returns the number of perfect groups of size size which are available in the library of finite perfect groups.

The purpose of the function is to provide a simple way to formulate a loop over all library groups of a given size.

SizeNumbersPerfectGroups( factor1, factor2 ... )

SizeNumbersPerfectGroups returns a list of the size numbers (see above) of all library groups that contain the specified factors among their composition factors. Each argument must either be the name of a simple group or an integer expression which is the product of the sizes of one or more cyclic factors. The function ignores the order in which the argmuments are given and, in fact, replaces any list of more than one integer expression among the arguments by their product.

The following text strings are accepted as simple group names.

•[] "A5", "A6", "A7", "A8", "A9" or "A(5)", "A(6)", "A(7)", "A(8)", "A(9)" for the alternating groups An, 5 ≤ n ≤ 9, \vspace-2mm

•[] "L2(q)" or "L(2,q)" for PSL(2,q), where q is any prime power with 4 ≤ q ≤ 125, \vspace-2mm

•[] "L3(q)" or "L(3,q)" for PSL(3,q) with 2 ≤ q ≤ 5, \vspace-2mm

•[] "U3(q)" or "U(3,q)" for PSU(2,q) with 3 ≤ q ≤ 5, \vspace-2mm

•[] "U4(2) or "U(4,2)" for PSU(4,2), \vspace-2mm

•[] "Sp4(4)" or "S(4,4)" for the symplectic group S(4,4),

•[] "Sz(8)" for the Suzuki group Sz(8), \vspace-2mm

•[] "M11", "M12", "M22" or "M(11)", "M(12)", "M(22)" for the Matthieu groups M11, M12, and M22, and

•[] "J1", "J2" or "J(1)", "J(2)" for the Janko groups J1 and J2. \vspace-2mm

Note that, for most of the groups, the preceding list offers two different names in order to be consistent with the notation used in HP89 as well as with the notation used in the DisplayCompositionSeries command of GAP3. However, as the names are compared as text strings, you are restricted to the above choice. Even expressions like "L2( 32 )" or "L2(2^5)" are not accepted.

As the use of the term PSU(n,q) is not unique in the literature, we state that here it denotes the factor group of SU(n,q) by its centre, where SU(n,q) is the group of all n × n unitary matrices with entries in GF(q2) and determinant 1.

The purpose of the function is to provide a simple way to formulate a loop over all library groups which contain certain composition factors.

DisplayInformationPerfectGroups( size )
DisplayInformationPerfectGroups( size, n )
DisplayInformationPerfectGroups( [ size, n ] )

DisplayInformationPerfectGroups displays some information about the library group G, say, which is specified by the size number [size,n] or by the two arguments size and n. If, in the second case, n is omitted, the function will loop over all library groups of size size.

The information provided for G includes the following items:

• a headline containing the size number [size,n] of G in the form size.n (the suffix .n will be suppressed if, up to isomorphism, G is the only perfect group of size size), \vspace-2mm

• a message if G is simple or quasisimple, i.e., if the factor group of G by its centre is simple, \vspace-2mm

• the description'' of the structure of G as it is given by Holt and Plesken in HP89 (see below), \vspace-2mm

• the size of the centre of G (suppressed, if G is simple),

• the prime decomposition of the size of G, \vspace-2mm

• orbit sizes for a faithful permutation representation of G which is provided by the library (see below), \vspace-2mm

• a reference to each occurrence of G in the tables of section 5.3 of HP89. Each of these references consists of a class number and an internal number (i,j) under which G is listed in that class. For some groups, there is more than one reference because these groups belong to more than one of the classes in the book.

Example:

    gap> DisplayInformationPerfectGroups( 30720, 3 );
#I Perfect group 30720.3:  A5 ( 2^4 E N 2^1 E 2^4 ) A
#I   centre = 1  size = 2^11*3*5  orbit size = 240
#I   Holt-Plesken class 1 (9,3)
gap> DisplayInformationPerfectGroups( 30720, 6 );
#I Perfect group 30720.6:  A5 ( 2^4 x 2^4 ) C N 2^1
#I   centre = 2  size = 2^11*3*5  orbit size = 384
#I   Holt-Plesken class 1 (9,6)
gap> DisplayInformationPerfectGroups( Factorial( 8 ) / 2 );
#I Perfect group 20160.1:  A5 x L3(2) 2^1
#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 5 + 16
#I   Holt-Plesken class 31 (1,1) (occurs also in class 32)
#I Perfect group 20160.2:  A5 2^1 x L3(2)
#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 7 + 24
#I   Holt-Plesken class 31 (1,2) (occurs also in class 32)
#I Perfect group 20160.3:  ( A5 x L3(2) ) 2^1
#I   centre = 2  size = 2^6*3^2*5*7  orbit size = 192
#I   Holt-Plesken class 31 (1,3)
#I Perfect group 20160.4:  simple group  A8
#I   size = 2^6*3^2*5*7  orbit size = 8
#I   Holt-Plesken class 26 (0,1)
#I Perfect group 20160.5:  simple group  L3(4)
#I   size = 2^6*3^2*5*7  orbit size = 21
#I   Holt-Plesken class 27 (0,1) 

For any library group G, the library files do not only provide a presentation, but, in addition, a list of one or more subgroups S1, ..., Sr of G such that there is a faithful permutation representation of G of degree i=1r G \!:\! Si on the set { Si g | 1 ≤ i ≤ r, g ∈ G } of the cosets of the Si. The respective permutation group is available via the PermGroup function described below. The DisplayInformationPerfectGroups function displays only the available degree. The message

 orbit size = 8

in the above example means that the available permutation representation is transitive and of degree 8, whereas the message

 orbit sizes = 7 + 24

means that a nontransitive permutation representation is available which acts on two orbits of size 7 and 24 respectively.

The notation used in the description'' of a group is explained in section 5.1.2 of HP89. We quote the respective page from there:

Within a class Q#p, an isomorphism type of groups will be denoted by an ordered pair of integers (r,n), where r ≥ 0 and n > 0. More precisely, the isomorphism types in Q # p of order pr \!\! | \!\! Q \!\! | will be denoted by (r,1), (r,2), (r,3), .... Thus Q will always get the size number (0,1).

In addition to the symbol (r,n), the groups in Q # p will also be given a more descriptive name. The purpose of this is to provide a very rough idea of the structure of the group. The names are derived in the following manner. First of all, the isomorphism classes of irreducible FpQ-modules M with | \!\! Q \!\! | | \!\! M \!\! | ≤ 106, where Fp is the field of order p, are assigned symbols. These will either be simply px, where x is the dimension of the module, or, if there is more than one isomorphism class of irreducible modules having the same dimension, they will be denoted by px, px', etc. The one-dimensional module with trivial Q-action will therefore be denoted by p1. These symbols will be listed under the description of Q. The group name consists essentially of a list of the composition factors working from the top of the group downwards; hence it always starts with the name of Q itself. (This convention is the most convenient in our context, but it is different from that adopted in the ATLAS (Conway {\it et al. 1985), for example, where composition factors are listed in the reverse order. For example, we denote a group isomorphic to SL(2,5) by A5 21 rather than 2 . A5.)

Some other symbols are used in the name, in order to give some idea of the relationship between these composition factors, and splitting properties. We shall now list these additional symbols.

•[×] between two factors denotes a direct product of FpQ-modules or groups. \vspace-2mm

•[C] (for commutator') between two factors means that the second lies in the commutator subgroup of the first. Similarly, a segment of the form (f1 \! × \! f2) C f3 would mean that the factors f1 and f2 commute modulo f3 and f3 lies in [f1,f2]. \vspace-2mm

•[A] (for abelian') between two factors indicates that the second is in the pth power (but not the commutator subgroup) of the first. A' may also follow the factors, if bracketed.

•[E] (for elementary abelian') between two factors indicates that together they generate an elementary abelian group (modulo subsequent factors), but that the resulting FpQ-module extension does not split. \vspace-2mm

•[N] (for nonsplit') before a factor indicates that Q (or possibly its covering group) splits down as far at this factor but not over the factor itself. So Q f1 N f2' means that the normal subgroup f1f2 of the group has no complement but, modulo f2, f1, does have a complement. \vspace-2mm

Brackets have their obvious meaning. Summarizing, we have:

•[×] = dirext product; \vspace-2mm

•[C] = commutator subgroup; \vspace-2mm

•[A] = abelian; \vspace-2mm

•[E] = elementary abelian; and \vspace-2mm

•[N] = nonsplit. \vspace-2mm Here are some examples.

•[(i)] A5 (24 E 21 E 24) A means that the pairs 24 E 21 and 21 E 24 are both elementary abelian of exponent 4. \vspace-1mm

•[(ii)] A5 (24 E 21 A) C 21 means that O2(G) is of symplectic type 21+5, with Frattini factor group of type 24 E 21. The A' after the 21 indicates that G has a central cyclic subgroup 21 A 21 of order 4. \vspace-1mm

•[(iii)] L3(2) ((21 E) \! × \! ( N 23 E 23' A) C) 23' means that the 23' factor at the bottom lies in the commutator subgroup of the pair 23 E 23' in the middle, but the lower pair 23' A 23' is abelian of exponent 4. There is also a submodule 21 E 23', and the covering group L3(2) 21 of L3(2) does not split over the 23 factor. (Since G is perfect, it goes without saying that the extension L3(2) 21 cannot split itself.) \vspace-2mm

We must stress that this notation does not always succeed in being precise or even unambiguous, and the reader is free to ignore it if it does not seem helpful.' }

If such a group description has been given in the book for G (and, in fact, this is the case for most of the library groups), it is displayed by the DisplayInformationPerfectGroups function. Otherwise the function provides a less explicit description of the (in these cases unique) Holt-Plesken class to which G belongs, together with a serial number if this is necessary to make it unique.

PerfectGroup( size )
PerfectGroup( size, n )
PerfectGroup( [ size, n ] )

PerfectGroup is the essential extraction function of the library. It returns a finitely presented group, G say, which is isomorphic to the library group specified by the size number [size,n] or by the two separate arguments size and n. In the second case, you may omit the parameter n. Then the default value is n = 1.

    gap> G := PerfectGroup( 6048 );
PerfectGroup(6048)
gap> G.generators;
[ a, b ]
gap> G.relators;
[ a^2, b^6, a*b*a*b*a*b*a*b*a*b*a*b*a*b,
a*b^2*a*b^2*a*b^2*a*b^-2*a*b^-2*a*b^-2,
a*b*a*b^-2*a*b*a*b^-2*a*b*a*b^-2*a*b*a*b^-1*a*b^-1 ]
gap> G.size;
6048
gap> G.description;
"U3(3)"
gap> G.subgroups;
[ Subgroup( PerfectGroup(6048), [ a, b*a*b*a*b*a*b^3 ] ) ] 

The generators and relators of G coincide with those given in HP89.

Note that, besides the components that are usually initialized for any finitely presented group, the group record of G contains the following components:

size:

the size of G,

isPerfect:

always true,

description:

the description of G as described with the DisplayInformationPerfectGroups function above,

source:

some internal information used by the library functions,

subgroups:

a list of subgroups S1, ..., Sr of G such that G acts faithfully on on the union of the sets of all cosets of the Si.

The last of these components exists only if G is one of the 488 nontrivial library groups which are given directly by a presentation on file, i.e., which are not constructed from other library groups in form of a direct, central, or subdirect product. It will be required by the following function.

PermGroup( G )

PermGroup returns a permutation group, P say, which is isomorphic to the given group G which is assumed to be a finitely presented perfect group that has been extracted from the library of finite perfect groups via the PerfectGroup function.

Let S1, ..., Sr be the subgroups listed in the component G.subgroups of the group record of G. Then the resulting group P is the permutation group of degree i=1r G \!:\! Si which is induced by G on the set { Si g | 1 ≤ i ≤ r, g ∈ G } of all cosets of the Si.

Example (continued):

    gap> P := PermGroup( G );
PermGroup(PerfectGroup(6048))
gap> P.size;
6048
gap> P.degree;
28 

Note that some of the library groups do not have a faithful permutation representation of small degree. Computations in these groups may be rather time consuming.

Example:

    gap> P := PermGroup( PerfectGroup( 129024, 2 ) );
PermGroup(PerfectGroup(129024,2))
gap> P.degree;
14336 

## 38.12 Irreducible Maximal Finite Integral Matrix Groups

A library of irreducible maximal finite integral matrix groups is provided with GAP3. It contains -class representatives for all of these groups of dimension at most 24, and -class representatives for those of dimension at most 11 or of dimension 13, 17, 19, or 23.

The groups provided in this library have been determined by Wilhelm Plesken, partially as joint work with Michael Pohst, or by members of his institute (Lehrstuhl B für Mathematik, RWTH Aachen). In particular, the data for the groups of dimensions 2 to 9 have been taken from the output of computer calculations which they performed in 1979 (see PP77, PP80). The -class representatives of the groups of dimension 10 have been determined and computed by Bernd Souvignier (Sou94), and those of dimensions 11, 13, and 17 have been recomputed for this library from the circulant Gram matrices given in Ple85, using the stand-alone programs for the computation of short vectors and Bravais groups which have been developed in Plesken's institute. The -class representatives of the groups of dimensions 19 and 23 had already been determined in Ple85. Gabriele Nebe has recomputed them for us. Her main contribution to this library, however, is that she has determined and computed the -class representatives of the groups of non-prime dimensions between 12 and 24 (see PN95, NP95, Neb95).

The library has been brought into GAP3 format by Volkmar Felsch. He has applied several GAP3 routines to check certain consistency of the data. However, the credit and responsibility for the lists remain with the authors. We are grateful to Wilhelm Plesken, Gabriele Nebe, and Bernd Souvignier for supplying their results to GAP3.

In the preceding acknowledgement, we used some notations that will also be needed in the sequel. We first define these.

Any integral matrix group G of dimension n is a subgroup of GLn(ℤ) as well as of GLn(ℚ) and hence lies in some conjugacy class of integral matrix groups under GLn(ℤ) and also in some conjugacy class of rational matrix groups under GLn(ℚ). As usual, we call these classes the -class and the -class of G, respectively. Note that any conjugacy class of subgroups of GLn(ℚ) contains at least one -class of subgroups of GLn(ℤ) and hence can be considered as the -class of some integral matrix group.

In the context of this library we are only concerned with -classes and -classes of subgroups of GLn(ℤ) which are irreducible and maximal finite in GLn(ℤ) (we will call them i.m.f. subgroups of GLn(ℤ)). We can distinguish two types of these groups:

First, there are those i.m.f. subgroups of GLn(ℤ) which are also maximal finite subgroups of GLn(ℚ). Let us denote the set of their -classes by Q1(n). It is clear from the above remark that Q1(n) just consists of the Q-classes of i.m.f. subgroups of GLn(ℚ).

Secondly, there is the set Q2(n) of the -classes of the remaining i.m.f. subgroups of GLn(ℤ), i.e., of those which are not maximal finite subgroups of GLn(ℚ). For any such group G, say, there is at least one class C ∈ Q1(n) such that G is conjugate under to a proper subgroup of some group H ∈ C. In fact, the class C is uniquely determined for any group G occurring in the library (though there seems to be no reason to assume that this property should hold in general). Hence we may call C the rational i.m.f. class of G. Finally, we will denote the number of classes in Q1(n) and Q2(n) by q1(n) and q2(n), respectively.

As an example, let us consider the case n = 4. There are 6 -classes of i.m.f. subgroups of GL4(ℤ) with representative subgroups G1, ..., G6 of isomorphsim types G1 ≅ W(F4), G2 ≅ D12 \wr C2, G3 ≅ G4 ≅ C2 × S5, G5 ≅ W(B4), and G6 ≅ (D12 \sf Y D12) \!:\! C2. The corresponding -classes, R1, ..., R6, say, are pairwise different except that R3 coincides with R4. The groups G1, G2, and G3 are i.m.f. subgroups of GL4(ℚ), but G5 and G6 are not because they are conjugate under GL4(ℚ) to proper subgroups of G1 and G2, respectively. So we have Q1(4) = { R1, R2, R3 }, Q2(4) = { R5, R6 }, q1(4) = 3, and q2(4) = 2.

The q1(n) -classes of i.m.f. subgroups of GLn(ℚ) have been determined for each dimension n ≤ 24. The current GAP3 library provides integral representative groups for all these classes. Moreover, all -classes of i.m.f. subgroups of GLn(ℤ) are known for n ≤ 11 and for n ∈ {13,17,19,23}. For these dimensions, the library offers integral representative groups for all -classes in Q1(n) and Q2(n) as well as for all -classes of i.m.f. subgroups of GLn(ℤ).

Any group G of dimension n given in the library is represented as the automorphism group G = Aut(F,L) = { g ∈ GLn(ℤ) | Lg = L and g F g tr = F } of a positive definite symmetric n × n matrix F ∈ ℤn × n on an n-dimensional lattice L ≅ ℤ1 × n (for details see e.g. PN95). GAP3 provides for G a list of matrix generators and the Gram matrix F.

The positive definite quadratic form defined by F defines a norm v F v tr for each vector v ∈ L, and there is only a finite set of vectors of minimal norm. These vectors are often simply called the short vectors''. Their set splits into orbits under G, and G being irreducible acts faithfully on each of these orbits by multiplication from the right. GAP3 provides for each of these orbits the orbit size and a representative vector.

Like most of the other GAP3 libraries, the library of i.m.f. integral matrix groups supplies an extraction function, ImfMatGroup. However, as the library involves only 423 different groups, there is no need for a selection or an example function. Instead, there are two functions, ImfInvariants and DisplayImfInvariants, which provide some -class invariants that can be extracted from the library without actually constructing the representative groups themselves. The difference between these two functions is that the latter one displays the resulting data in some easily readable format, whereas the first one returns them as record components so that you can properly access them.

We shall give an individual description of each of the library functions, but first we would like to insert a short remark concerning their names: Any self-explaining name of a function handling irreducible maximal finite integral matrix groups would have to include this term in full length and hence would grow extremely long. Therefore we have decided to use the abbreviation Imf instead in order to restrict the names to some reasonable length.

The first three functions can be used to formulate loops over the classes.

ImfNumberQQClasses( dim )
ImfNumberQClasses( dim )
ImfNumberZClasses( dim, q )

ImfNumberQQClasses returns the number q1(dim) of -classes of i.m.f. rational matrix groups of dimension dim. Valid values of dim are all positive integers up to 24.

Note: In order to enable you to loop just over the classes belonging to Q1(dim), we have arranged the list of -classes of dimension dim for any dimension dim in the library such that, whenever the classes of Q2(dim) are known, too, i.e., in the cases dim ≤ 11 or dim ∈ {13,17,19,23}, the classes of Q1(dim) precede those of Q2(dim) and hence are numbered from 1 to q1(dim).

ImfNumberQClasses returns the number of -classes of groups of dimension dim which are available in the library. If dim ≤ 11 or dim ∈ {13,17,19,23}, this is the number q1(dim) + q2(dim) of -classes of i.m.f. subgroups of GLdim(ℤ). Otherwise, it is just the number q1(dim) of -classes of i.m.f. subgroups of GLdim(ℚ). Valid values of dim are all positive integers up to 24.

ImfNumberZClasses returns the number of -classes in the q th -class of i.m.f. integral matrix groups of dimension dim. Valid values of dim are all positive integers up to 11 and all primes up to 23.

DisplayImfInvariants( dim, q )
DisplayImfInvariants( dim, q, z )

DisplayImfInvariants displays the following -class invariants of the groups in the z th -class in the q th -class of i.m.f. integral matrix groups of dimension dim:

• its -class number in the form dim.q.z, if dim is at most 11 or a prime, or its -class number in the form dim.q, else, \vspace-2mm

• a message if the group is solvable, \vspace-2mm

• the size of the group, \vspace-2mm

• the isomorphism type of the group, \vspace-2mm

• the elementary divisors of the associated quadratic form,

• the sizes of the orbits of short vectors (these sizes are the degrees of the faithful permutation representations which you may construct using the PermGroup or PermGroupImfGroup commands below), \vspace-2mm

• the norm of the associated short vectors, \vspace-2mm

• only in case that the group is not an i.m.f. group in GLn(ℚ): an appropriate message, including the -class number of the corresponding rational i.m.f. class.

If you specify the value 0 for any of the parameters dim, q, or z, the command will loop over all available dimensions, -classes of given dimension, or -classes within the given -class, respectively. Otherwise, the values of the arguments must be in range. A value z ≠ 1 must not be specified if the -classes are not known for the given dimension, i.e., if dim > 11 and dim ∉ {13,17,19,23}. The default value of z is 1. This value of z will be accepted even if the -classes are not known. Then it specifies the only representative group which is available for the q th -class. The greatest legal value of dim is 24.

    gap> DisplayImfInvariants( 3, 1, 0 );
#I Z-class 3.1.1:  Solvable, size = 2^4*3
#I   isomorphism type = C2 wr S3 = C2 x S4 = W(B3)
#I   elementary divisors = 1^3
#I   orbit size = 6, minimal norm = 1
#I Z-class 3.1.2:  Solvable, size = 2^4*3
#I   isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I   elementary divisors = 1*4^2
#I   orbit size = 8, minimal norm = 3
#I Z-class 3.1.3:  Solvable, size = 2^4*3
#I   isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I   elementary divisors = 1^2*4
#I   orbit size = 12, minimal norm = 2
gap> DisplayImfInvariants( 8, 15, 1 );
#I Z-class 8.15.1:  Solvable, size = 2^5*3^4
#I   isomorphism type = C2 x (S3 wr S3)
#I   elementary divisors = 1*3^3*9^3*27
#I   orbit size = 54, minimal norm = 8
#I   not maximal finite in GL(8,Q), rational imf class is 8.5
gap> DisplayImfInvariants( 20, 23 );
#I Q-class 20.23:  Size = 2^5*3^2*5*11
#I   isomorphism type = (PSL(2,11) x D12).C2
#I   elementary divisors = 1^18*11^2
#I   orbit size = 3*660 + 2*1980 + 2640 + 3960, minimal norm = 4 

Note that the DisplayImfInvariants function uses a kind of shorthand to display the elementary divisors. E. g., the expression 1*3^3*9^3*27 in the preceding example stands for the elementary divisors 1,3,3,3,9,9,9,27. (See also the next example which shows that the ImfInvariants function provides the elementary divisors in form of an ordinary GAP3 list.)

In the description of the isomorphism types the following notations are used:

•[] \beginitemize

•[] \beginitemize

•[{\makebox[15mm][l]AxB}] denotes a direct product of a group A by a group B, \vspace-1mm

•[{\makebox[15mm][l]AsubdB}] denotes a subdirect product of A by B, \vspace-1mm

•[{\makebox[15mm][l]AYB}] denotes a central product of A by B, \vspace-1mm

•[{\makebox[15mm][l]AwrB}] denotes a wreath product of A by B, \vspace-1mm

•[{\makebox[15mm][l]A:B}] denotes a split extension of A by B, \vspace-1mm

•[{\makebox[15mm][l]A.B}] denotes just an extension of A by B (split or nonsplit). \vspace-2mm \enditemize \enditemize The groups involved are

• the cyclic groups Cn, dihedral groups Dn, and generalized quaternion groups Qn of order n, denoted by Cn, Dn, and Qn, respectively, \vspace-2mm

• the alternating groups An and symmetric groups Sn of degree n, denoted by An and Sn, respectively, \vspace-2mm

• the linear groups GLn(q), PGLn(q), SLn(q), and PSLn(q), denoted by GL(n,q), PGL(n,q), SL(n,q), and PSL(n,q), respectively,

• the unitary groups SUn(q) and PSUn(q), denoted by SU(n,q) and PSU(n,q), respectively,

• the symplectic groups Sp(n,q), denoted by Sp(n,q),

• the orthogonal group O8+(2), denoted by O+(8,2),

• the extraspecial groups 2+1+8, 3+1+2, 3+1+4, and 5+1+2, denoted by 2+^(1+8), 3+^(1+2), 3+^(1+4), and 5+^(1+2), respectively, \vspace-2mm

• the Chevalley group G2(3), denoted by G(2,3),

• the Weyl groups W(An), W(Bn), W(Dn), W(En), and W(F4), denoted by W(An), W(Bn), W(Dn), W(En), and W(F4), respectively, \vspace-2mm

• the sporadic simple groups Co1, Co2, Co3, HS, J2, M12, M22, M23, M24, and Mc, denoted by Co1, Co2, Co3, HS, J2, M12, M22, M23, M24, and Mc, respectively, \vspace-2mm

• a point stabilizer of index 11 in M11, denoted by M10.

As mentioned above, the data assembled by the DisplayImfInvariants command are cheap data'' in the sense that they can be provided by the library without loading any of its large matrix files or performing any matrix calculations. The following function allows you to get proper access to these cheap data instead of just displaying them.

ImfInvariants( dim, q )
ImfInvariants( dim, q, z )

ImfInvariants returns a record which provides some -class invariants of the groups in the z th -class in the q th -class of i.m.f. integral matrix groups of dimension dim. A value z ≠ 1 must not be specified if the -classes are not known for the given dimension, i.e., if dim > 11 and dim ∉ {13,17,19,23}. The default value of z is 1. This value of z will be accepted even if the -classes are not known. Then it specifies the only representative group which is available for the q th -class. The greatest legal value of dim is 24.

The resulting record contains six or seven components:

size:

the size of any representative group G,

isSolvable:

is true if G is solvable,

isomorphismType:

a text string describing the isomorphism type of G (in the same notation as used by the DisplayImfInvariants command above),

elementaryDivisors:

the elementary divisors of the associated Gram matrix F (in the same format as the result of the ElementaryDivisorsMat function, see ElementaryDivisorsMat),

minimalNorm:

the norm of the associated short vectors,

sizesOrbitsShortVectors:

the sizes of the orbits of short vectors under F,

maximalQClass:

the -class number of an i.m.f. group in GLn(ℚ) that contains G as a subgroup (only in case that not G itself is an i.m.f. subgroup of GLn(ℚ)).

Note that four of these data, namely the group size, the solvability, the isomorphism type, and the corresponding rational i.m.f. class, are not only -class invariants, but also -class invariants.

Note further that, though the isomorphism type is a -class invariant, you will sometimes get different descriptions for different -classes of the same -class (as, e.g., for the classes 3.1.1 and 3.1.2 in the last example above). The purpose of this behaviour is to provide some more information about the underlying lattices.

    gap> ImfInvariants( 8, 15, 1 );
rec(
size := 2592,
isSolvable := true,
isomorphismType := "C2 x (S3 wr S3)",
elementaryDivisors := [ 1, 3, 3, 3, 9, 9, 9, 27 ],
minimalNorm := 8,
sizesOrbitsShortVectors := [ 54 ],
maximalQClass := 5 )
gap> ImfInvariants( 24, 1 ).size;
10409396852733332453861621760000
gap> ImfInvariants( 23, 5, 2 ).sizesOrbitsShortVectors;
[ 552, 53130 ]
gap> for i in [ 1 .. ImfNumberQClasses( 22 ) ] do
>    Print( ImfInvariants( 22, i ).isomorphismType, "\n" ); od;
C2 wr S22 = W(B22)
(C2 x PSU(6,2)).S3
(C2 x S3) wr S11 = (C2 x W(A2)) wr S11
(C2 x S12) wr C2 = (C2 x W(A11)) wr C2
C2 x S3 x S12 = C2 x W(A2) x W(A11)
(C2 x HS).C2
(C2 x Mc).C2
C2 x S23 = C2 x W(A22)
C2 x PSL(2,23)
C2 x PSL(2,23)
C2 x PGL(2,23)
C2 x PGL(2,23) 

ImfMatGroup( dim, q )
ImfMatGroup( dim, q, z )

ImfMatGroup is the essential extraction function of this library. It returns a representative group, G say, of the z th -class in the q th -class of i.m.f. integral matrix groups of dimension dim. A value z ≠ 1 must not be specified if the -classes are not known for the given dimension, i.e., if dim > 11 and dim ∉ {13,17,19,23}. The default value of z is 1. This value of z will be accepted even if the -classes are not known. Then it specifies the only representative group which is available for the q th -class. The greatest legal value of dim is 24.

    gap> G := ImfMatGroup( 5, 1, 3 );
ImfMatGroup(5,1,3)
gap> for m in G.generators do PrintArray( m ); od;
[ [  -1,   0,   0,   0,   0 ],
[   0,   1,   0,   0,   0 ],
[   0,   0,   0,   1,   0 ],
[  -1,  -1,  -1,  -1,   2 ],
[  -1,   0,   0,   0,   1 ] ]
[ [  0,  1,  0,  0,  0 ],
[  0,  0,  1,  0,  0 ],
[  0,  0,  0,  1,  0 ],
[  1,  0,  0,  0,  0 ],
[  0,  0,  0,  0,  1 ] ] 

The group record of G contains the usual components of a matrix group record. In addition, it includes the same six or seven records as the resulting record of the ImfInvariants function described above, namely the components size, isSolvable, isomorphismType, elementaryDivisors, minimalNorm, and sizesOrbitsShortVectors and, if G is not a rational i.m.f. group, maximalQClass. Moreover, there are the two components

form:

the associated Gram matrix F,

repsOrbitsShortVectors:

representatives of the orbits of short vectors under F.

The last of these components will be required by the PermGroup function below.

Example:

    gap> G.size;
3840
gap> G.isomorphismType;
"C2 wr S5 = C2 x W(D5)"
gap> PrintArray( G.form );
[ [  4,  0,  0,  0,  2 ],
[  0,  4,  0,  0,  2 ],
[  0,  0,  4,  0,  2 ],
[  0,  0,  0,  4,  2 ],
[  2,  2,  2,  2,  5 ] ]
gap> G.elementaryDivisors;
[ 1, 4, 4, 4, 4 ]
gap> G.minimalNorm;
4 

If you want to perform calculations in such a matrix group G you should be aware of the fact that GAP3 offers much more efficient permutation group routines than matrix group routines. Hence we recommend that you do your computations, whenever it is possible, in the isomorphic permutation group that is induced by the action of G on one of the orbits of the associated short vectors. You may call one of the following functions to get such a permutation group.

PermGroup( G )

PermGroup returns the permutation group which is induced by the given i.m.f. integral matrix group G on an orbit of minimal size of G on the set of short vectors (see also PermGroupImfGroup below).

The permutation representation is computed by first constructing the specified orbit, S say, of short vectors and then computing the permutations which are induced on S by the generators of G. We would like to warn you that in case of a large orbit this procedure may be space and time consuming. Fortunately, there are only five -classes in the library for which the smallest orbit of short vectors is of size greater than 20000, the worst case being the orbit of size 196560 for the Leech lattice (dim = 24, q = 3).

As mentioned above, PermGroup constructs the required permutation group, P say, as the image of G under the isomorphism between the matrices in G and their action on S. Moreover, it constructs the inverse isomorphism from P to G, φ say, and returns it in the group record component P.bijection of P. In fact, φ is constructed by determining a -base B ⊂ S of 1 × dim in S and, in addition, the associated base change matrix M which transforms B into the standard base of 1 × dim. Then the image φ(p) of any permutation p ∈ P can be easily computed: If, for 1 ≤ i ≤ dim, bi is the position number in S of the i th base vector in B, it suffices to look up the vector whose position number in S is the image of bi under p and to multiply this vector by M to get the i th row of φ(p).

You may use φ at any time to compute the images in G of permutations in P or to compute the preimages in P of matrices in G.

The record of P contains, in addition to the usual components of permutation group records, some special components. The most important of those are:

isomorphismType:

a text string describing the isomorphism type of P (in the same notation as used by the DisplayImfInvariants command above),

matGroup:

the associated matrix group G,

bijection:

the isomorphism φ from P to G,

orbitShortVectors:

the orbit S of short vectors (needed for φ),

baseVectorPositions:

the position numbers of the base vectors in B with respect to S (needed for φ),

baseChangeMatrix:

the base change matrix M (needed for φ).

As an example, let us compute a set R of matrix generators for the solvable residuum of the group G that we have constructed in the preceding example.

    gap> # Perform the computations in an isomorphic permutation group.
gap> P := PermGroup( G );
PermGroup(ImfMatGroup(5,1,3))
gap> P.generators;
[ ( 1, 7, 6)( 2, 9)( 4, 5,10), ( 2, 3, 4, 5)( 6, 9, 8, 7) ]
gap> D := DerivedSubgroup( P );
Subgroup( PermGroup(ImfMatGroup(5,1,3)),
[ ( 1, 2,10, 9)( 3, 8)( 4, 5)( 6, 7),
( 1, 6)( 2, 7, 9, 4)( 3, 8)( 5,10), ( 1, 5,10, 6)( 2, 8, 9, 3) ] )
gap> Size( D );
960
gap> IsPerfect( D );
true
gap> # Now move the results back to the matrix group.
gap> phi := P.bijection;;
gap> R := List( D.generators, p -> Image( phi, p ) );;
gap> for m in R do PrintArray( m ); od;
[ [  -1,  -1,  -1,  -1,   2 ],
[   0,  -1,   0,   0,   0 ],
[   0,   0,   0,   1,   0 ],
[   0,   0,   1,   0,   0 ],
[  -1,  -1,   0,   0,   1 ] ]
[ [   0,   0,  -1,   0,   0 ],
[   0,  -1,   0,   0,   0 ],
[   1,   0,   0,   0,   0 ],
[  -1,  -1,  -1,  -1,   2 ],
[   0,  -1,  -1,   0,   1 ] ]
[ [   0,  -1,   0,   0,   0 ],
[   1,   0,   0,   0,   0 ],
[   0,   0,   1,   0,   0 ],
[  -1,  -1,  -1,  -1,   2 ],
[   0,  -1,   0,  -1,   1 ] ]
gap> # The PreImage function allows us to use the inverse of phi.
gap> PreImage( phi, R ) = D.generators;
true 

PermGroupImfGroup( G, n )

PermGroupImfGroup returns the permutation group which is induced by the given i.m.f. integral matrix group G on its n th orbit of short vectors. The only difference to the above PermGroup function is that you can specify the orbit to be used. In fact, as the orbits of short vectors are sorted by increasing sizes, the function PermGroup( G ) has been implemented such that it is equivalent to PermGroupImfGroup( G, 1 ).

    gap> ImfInvariants( 12, 9 ).sizesOrbitsShortVectors;
[ 120, 300 ]
gap> G := ImfMatGroup( 12, 9 );
ImfMatGroup(12,9)
gap> P1 := PermGroupImfGroup( G, 1 );
PermGroup(ImfMatGroup(12,9))
gap> P1.degree;
120
gap> P2 := PermGroupImfGroup( G, 2 );
PermGroupImfGroup(ImfMatGroup(12,9),2)
gap> P2.degree;
300 

## 38.13 The Crystallographic Groups Library

GAP3 provides a library of crystallographic groups of dimensions 2, 3, and 4 which covers many of the data that have been listed in the book Crystallographic groups of four-dimensional space'' BBNWZ78. It has been brought into GAP3 format by Volkmar Felsch.

How to access the data of the book

Among others, the library offers functions which provide access to the data listed in the Tables 1, 5, and 6 of BBNWZ78:

• The information on the crystal families listed in Table 1 can be reproduced using the DisplayCrystalFamily function.

• Similarly, the DisplayCrystalSystem function can be used to reproduce the information on the crystal systems provided in Table 1.

• The information given in the -class headlines of Table 1 can be displayed by the DisplayQClass function, whereas the FpGroupQClass function can be used to reproduce the presentations that are listed in Table 1 for the -class representatives.

• The information given in the -class headlines of Table 1 will be covered by the results of the DisplayZClass function, and the matrix generators of the -class representatives can be constructed by calling the MatGroupZClass function.

• The DisplaySpaceGroupType and the DisplaySpaceGroupGenerators functions can be used to reproduce all of the information on the space-group types that is provided in Table 1.

• The normalizers listed in Table 5 can be reproduced by calling the NormalizerZClass function.

• Finally, the CharTableQClass function will compute the character tables listed in Table 6, whereas the isomorphism types given in Table 6 may be obtained by calling the DisplayQClass function.

The display functions mentioned in the above list print their output with different indentation. So, calling them in a suitably nested loop, you may produce a listing in which the information about the objects of different type will be properly indented as has been done in Table 1 of BBNWZ78.

Representation of space groups in {\GAP}

Probably the most important function in the library is the SpaceGroup function which provides representatives of the affine classes of space groups. A space group of dimension n is represented by an (n+1)-dimensional rational matrix group as follows.

If S is an n-dimensional space group, then each element α ∈ S is an affine mapping α\!: V → V of an n-dimensional -vector space V onto itself. Hence α can be written as the sum of an appropriate invertible linear mapping φ\!: V → V and a translation by some translation vector t ∈ V such that, if we write mappings from the left, we have α(v) = φ(v) + t for all v ∈ V.

If we fix a basis of V and then replace each v ∈ V by the column vector of its coefficients with respect to that basis (and hence V by the isomorphic column vector space n × 1), we can describe the linear mapping φ involved in α by an n × n matrix Mφ ∈ GLn(ℝ) which acts by multiplication from the left on the column vectors in n × 1. Hence, if we identify V with n × 1, we have α(v) = Mφ . v + t for all v ∈ ℝn × 1.

Moreover, if we extend each column vector v ∈ ℝn × 1 to a column [ \! \! \catcode|=12 \begintabularc \catcode|=13 v \cr 1 \endtabular \! \! ] of length n+1 by adding an entry 1 in the last position and if we define an (n+1) × (n+1) matrix Mα = [ \catcode|=12 \begintabular{cc} \catcode\=13 M_φ & t \cr \hline 0 & 1 \endtabular ], we have [ \! \! \catcode|=12 \begintabularc \catcode|=13 α(v) \cr 1 \endtabular \! \! ] = Mα . [ \! \! \catcode|=12 \begintabularc \catcode|=13 v \cr 1 \endtabular \! \! ] for all v ∈ ℝn × 1. This means that we can represent the space group S by the isomorphic group M(S) = { Mα | α ∈ S }. The submatrices Mφ occurring in the elements of M(S) form an n × n matrix group P(S), the point group'' of M(S). In fact, we can choose the basis of n × 1 such that Mφ ∈ GLn(ℤ) and t ∈ ℚn × 1 for all Mα ∈ M(S). In particular, the space group representatives that are normally used by the crystallographers are of this form, and the book BBNWZ78 uses the same convention.

Before we describe all available library functions in detail, we have to add three remarks.

Remark 1

The concepts used in this section are defined in chapter 1 (Basic definitions) of BBNWZ78. However, note that the definition of the concept of a crystal system given on page 16 of that book relies on the following statement about -classes:

•[] For a -class C there is a unique holohedry H such that each f.u. group in C is a subgroup of some f.u. group in H, but is not a subgroup of any f.u. group belonging to a holohedry of smaller order.

This statement is correct for dimensions 1, 2, 3, and 4, and hence the definition of crystal system'' given on page 16 of BBNWZ78 is known to be unambiguous for these dimensions. However, there is a counterexample to this statement in seven-dimensional space so that the definition breaks down for some higher dimensions.

Therefore, the authors of the book have since proposed to replace this definition of crystal system'' by the following much simpler one, which has been discussed in more detail in NPW81. To formulate it, we use the intersections of -classes and Bravais flocks as introduced on page 17 of BBNWZ78, and we define the classification of the set of all -classes into crystal systems as follows:

•[] Definition: A crystal system (introduced as an equivalence class of -classes) consists of full geometric crystal classes. The -classes of two (geometric) crystal classes belong to the same crystal system if and only if these geometric crystal classes intersect the same set of Bravais flocks of -classes.

>From this definition of a crystal system of -classes one then obtains crystal systems of f.u. groups, of space-group types, and of space groups in the same manner as with the preceding definitions in the book.

The new definition is unambiguous for all dimensions. Moreover, it can be checked from the tables in the book that it defines the same classification as the old one for dimensions 1, 2, 3, and 4.

It should be noted that the concept of crystal family is well-defined independently of the dimension if one uses the more natural'' second definition of it at the end of page 17. Moreover, the first definition of crystal family on page 17 defines the same concept as the second one if the now proposed definition of crystal system is used.

Remark 2

The second remark just concerns a different terminology in the tables of BBNWZ78 and in the current library. In group theory, the number of elements of a finite group usually is called the order'' of the group. This notation has been used throughout in the book. Here, however, we will follow the GAP3 conventions and use the term size'' instead.

Remark 3

The third remark concerns a problem in the use of the space groups that should be well understood.

There is an alternative to the representation of the space group elements by matrices of the form [ \catcode|=12 \begintabular{c

c}
\catcode\
=13 M_φ & t \cr \hline 0 & 1 \endtabular ] as described above. Instead of considering the coefficient vectors as columns we may consider them as rows. Then we can associate to each affine mapping α ∈ S an (n+1) × (n+1) matrix Mα = [ \catcode|=12 \begintabular{c
c}
\catcode\
=13
\overlineM_{\overlineφ} & 0 \cr \hline \overlinet & 1 \endtabular ] with Mφ ∈ GLn(ℝ) and t ∈ ℝ1 × n such that [α(v),1] = [v,1] . Mα for all v ∈ ℝ1 × n, and we may represent S by the matrix group M(S) = { Mα | α ∈ S }. Again, we can choose the basis of 1 × n such that Mφ ∈ GLn(ℤ) and t ∈ ℚ1 × n for all MαM(S).

From the mathematical point of view, both approaches are equivalent. In particular, M(S) and M(S) are isomorphic, for instance via the isomorphism τ mapping Mα ∈ M(S) to (Mα tr)-1. Unfortunately, however, neither of the two is a good choice for our GAP3 library.

The first convention, using matrices which act on column vectors from the left, is not consistent with the fact that actions in GAP3 are usually from the right.

On the other hand, if we choose the second convention, we run into a problem with the names of the space groups as introduced in BBNWZ78. Any such name does not just describe the abstract isomorphism type of the respective space group S, but reflects properties of the matrix group M(S). In particular, it contains as a leading part the name of the -class of the associated point group P(S). Since the classification of space groups by affine equivalence is tantamount to their classification by abstract isomorphism, M(S) lies in the same affine class as M(S) and hence should get the same name as M(S). But the point group P(S) that occurs in that name is not always -equivalent to the point group P(S) of M(S). For example, the isomorphism τ\!\!: M(S) → M(S) defined above maps the -class representative with the parameters [3,7,3,2] (in the notation described below) to the -class representative with the parameters [3,7,3,3]. In other words: The space group names introduced for the groups M(S) in BBNWZ78 lead to confusing inconsistencies if assigned to the groups M(S).

In order to avoid this confusion we decided that the first convention is the lesser evil. So the GAP3 library follows the book, and if you call the SpaceGroup function you will get the same space group representatives as given there. This does not cause any problems as long as you do calculations within these groups treating them just as matrix groups of certain isomorphism types. However, if it is necesary to consider the action of a space group as affine mappings on the natural lattice, you need to use the transposed representation of the space group. For this purpose the library offers the TransposedSpaceGroup function which not only transposes the matrices, but also adapts appropriately the associated group presentation.

Both these functions are described in detail in the following.

The library functions

NrCrystalFamilies( dim )

NrCrystalFamilies returns the number of crystal families in case of dimension dim. It can be used to formulate loops over the crystal families.

There are 4, 6, and 23 crystal families of dimension 2, 3, and 4, respectively.

    gap> n := NrCrystalFamilies( 4 );
23 

DisplayCrystalFamily( dim, family )

DisplayCrystalFamily displays for the specified crystal family essentially the same information as is provided for that family in Table 1 of BBNWZ78, namely

• the family name,

• the number of parameters,

• the common rational decomposition pattern,

• the common real decomposition pattern,

• the number of crystal systems in the family, and

• the number of Bravais flocks in the family.

For details see BBNWZ78.

    gap> DisplayCrystalFamily( 4, 17 );
#I Family XVII: cubic orthogonal; 2 free parameters;
#I  Q-decomposition pattern 1+3; R-decomposition pattern 1+3;
#I  2 crystal systems; 6 Bravais flocks
gap> DisplayCrystalFamily( 4, 18 );
#I Family XVIII: octagonal; 2 free parameters;
#I  Q-irreducible; R-decomposition pattern 2+2;
#I  1 crystal system; 1 Bravais flock
gap> DisplayCrystalFamily( 4, 21 );
#I Family XXI: di-isohexagonal orthogonal; 1 free parameter;
#I  R-irreducible; 2 crystal systems; 2 Bravais flocks 

NrCrystalSystems( dim )

NrCrystalSystems returns the number of crystal systems in case of dimension dim. It can be used to formulate loops over the crystal systems.

There are 4, 7, and 33 crystal systems of dimension 2, 3, and 4, respectively.

    gap> n := NrCrystalSystems( 2 );
4 

The following two functions are functions of crystal systems.

Each crystal system is characterized by a pair (dim,\,system) where dim is the associated dimension, and system is the number of the crystal system.

DisplayCrystalSystem( dim, system )

DisplayCrystalSystem displays for the specified crystal system essentially the same information as is provided for that system in Table 1 of BBNWZ78, namely

• the number of -classes in the crystal system and

• the identification number, i.e., the tripel (dim,\,system,\,q-class) described below, of the -class that is the holohedry of the crystal system.

For details see BBNWZ78.

    gap> for sys in [ 1 .. 4 ] do  DisplayCrystalSystem( 2, sys );  od;
#I  Crystal system 1: 2 Q-classes; holohedry (2,1,2)
#I  Crystal system 2: 2 Q-classes; holohedry (2,2,2)
#I  Crystal system 3: 2 Q-classes; holohedry (2,3,2)
#I  Crystal system 4: 4 Q-classes; holohedry (2,4,4) 

NrQClassesCrystalSystem( dim, system )

NrQClassesCrystalSystem returns the number of -classes within the given crystal system. It can be used to formulate loops over the -classes.

The following five functions are functions of -classes.

In general, the parameters characterizing a -class will form a triple (dim,\,system,\,q-class) where dim is the associated dimension, system is the number of the associated crystal system, and q-class is the number of the -class within the crystal system. However, in case of dimensions 2 or 3, a -class may also be characterized by a pair (dim, IT-number) where IT-number is the number in the International Tables for Crystallography Hah83 of any space-group type lying in (a -class of) that -class, or just by the Hermann-Mauguin symbol of any space-group type lying in (a -class of) that -class.

The Hermann-Mauguin symbols which we use in GAP3 are the short Hermann-Mauguin symbols defined in the 1983 edition of the International Tables Hah83, but any occurring indices are expressed by ordinary integers, and bars are replaced by minus signs. For example, the Hermann-Mauguin symbol P421m will be represented by the string "P-421m".

DisplayQClass( dim, system, q-class )
DisplayQClass( dim, IT-number )
DisplayQClass( Hermann-Mauguin-symbol )

DisplayQClass displays for the specified -class essentially the same information as is provided for that -class in Table 1 of BBNWZ78 (except for the defining relations given there), namely

• the size of the groups in the -class,

• the isomorphism type of the groups in the -class,

• the Hurley pattern,

• the rational constituents,

• the number of -classes in the -class, and

• the number of space-group types in the -class.

For details see BBNWZ78.

    gap> DisplayQClass( "p2" );
#I   Q-class H (2,1,2): size 2; isomorphism type 2.1 = C2;
#I    Q-constituents 2*(2,1,2); cc; 1 Z-class; 1 space group
gap> DisplayQClass( "R-3" );
#I   Q-class (3,5,2): size 6; isomorphism type 6.1 = C6;
#I    Q-constituents (3,1,2)+(3,4,3); ncc; 2 Z-classes; 2 space grps
gap> DisplayQClass( 3, 195 );
#I   Q-class (3,7,1): size 12; isomorphism type 12.5 = A4;
#I    C-irreducible; 3 Z-classes; 5 space grps
gap> DisplayQClass( 4, 27, 4 );
#I   Q-class H (4,27,4): size 20; isomorphism type 20.3 = D10xC2;
#I    Q-irreducible; 1 Z-class; 1 space group
gap> DisplayQClass( 4, 29, 1 );
#I  *Q-class (4,29,1): size 18; isomorphism type 18.3 = D6xC3;
#I    R-irreducible; 3 Z-classes; 5 space grps 

Note in the preceding examples that, as pointed out above, the term size'' denotes the order of a representative group of the specified -class and, of course, not the (infinite) class length.

FpGroupQClass( dim, system, q-class )
FpGroupQClass( dim, IT-number )
FpGroupQClass( Hermann-Mauguin-symbol )

FpGroupQClass returns a finitely presented group F, say, which is isomorphic to the groups in the specified -class.

The presentation of that group is the same as the corresponding presentation given in Table 1 of BBNWZ78 except for the fact that its generators are listed in reverse order. The reason for this change is that, whenever the group in question is solvable, the resulting generators form an AG system (as defined in GAP3) if they are numbered from the top to the bottom'', and the presentation is a polycyclic power commutator presentation. The AgGroupQClass function described next will make use of this fact in order to construct an ag group isomorphic to F.

Note that, for any -class in the specified -class, the matrix group returned by the MatGroupZClass function (see below) not only is isomorphic to F, but also its generators satisfy the defining relators of F.

Besides of the usual components, the group record of F will have an additional component F.crQClass which saves a list of the parameters that specify the given -class.

    gap> F := FpGroupQClass( 4, 20, 3 );
FpGroupQClass( 4, 20, 3 )
gap> F.generators;
[ f.1, f.2 ]
gap> F.relators;
[ f.1^2*f.2^-3, f.2^6, f.2^-1*f.1^-1*f.2*f.1*f.2^-4 ]
gap> F.size;
12
gap> F.crQClass;
[ 4, 20, 3 ] 

AgGroupQClass( dim, system, q-class )
AgGroupQClass( dim, IT-number )
AgGroupQClass( Hermann-Mauguin-symbol )

AgGroupQClass returns an ag group A, say, isomorphic to the groups in the specified -class, if these groups are solvable, or the value false (together with an appropriate warning), otherwise.

A is constructed by first establishing a finitely presented group (as it would be returned by the FpGroupQClass function described above) and then constructing from it an isomorphic ag group. If the underlying AG system is not yet a PAG system (see sections More about Ag Words and More about Ag Groups), it will be refined appropriately (and a warning will be displayed).

Besides of the usual components, the group record of A will have an additional component A.crQClass which saves a list of the parameters that specify the given -class.

    gap> A := AgGroupQClass( 4, 31, 3 );
#I  Warning: a non-solvable group can't be represented as an ag group
false
gap> A := AgGroupQClass( 4, 20, 3 );
#I  Warning: the presentation has been extended to get a PAG system
AgGroupQClass( 4, 20, 3 )
gap> A.generators;
[ f.1, f.21, f.22 ]
gap> A.size;
12
gap> A.crQClass;
[ 4, 20, 3 ] 

CharTableQClass( dim, system, q-class )
CharTableQClass( dim, IT-number )
CharTableQClass( Hermann-Mauguin-symbol )

CharTableQClass returns the character table T, say, of a representative group of (a -class of) the specified -class.

Although the set of characters can be considered as an invariant of the specified -class, the resulting table will depend on the order in which GAP3 sorts the conjugacy classes of elements and the irreducible characters and hence, in general, will not coincide with the corresponding table presented in BBNWZ78.

CharTableQClass proceeds as follows. If the groups in the given -class are solvable, then it first calls the AgGroupQClass and RefinedAgSeries functions to get an isomorphic ag group with a PAG system, and then it calls the CharTable function to compute the character table of that ag group. In the case of one of the five -classes of dimension 4 whose groups are not solvable, it first calls the FpGroupQClass function to get an isomorphic finitely presented group, then it constructs a specially chosen faithful permutation representation of low degree for that group, and finally it determines the character table of the resulting permutation group again by calling the CharTable function.

In general, the above strategy will be much more efficient than the alternative possibilities of calling the CharTable function for a finitely presented group provided by the FpGroupQClass function or for a matrix group provided by the MatGroupZClass function.

    gap> T := CharTableQClass( 4, 20, 3 );;
gap> DisplayCharTable( T );
CharTableQClass( 4, 20, 3 )

2  2  1  1  2  2  2
3  1  1  1  1  .  .

1a 3a 6a 2a 4a 4b
2P 1a 3a 3a 1a 2a 2a
3P 1a 1a 2a 2a 4b 4a
5P 1a 3a 6a 2a 4a 4b

X.1     1  1  1  1  1  1
X.2     1  1  1  1 -1 -1
X.3     1  1 -1 -1  A -A
X.4     1  1 -1 -1 -A  A
X.5     2 -1  1 -2  .  .
X.6     2 -1 -1  2  .  .

A = E(4)
= ER(-1) = i 

NrZClassesQClass( dim, system, q-class )
NrZClassesQClass( dim, IT-number )
NrZClassesQClass( Hermann-Mauguin-symbol )

NrZClassesQClass returns the number of -classes within the given -class. It can be used to formulate loops over the -classes.

The following functions are functions of -classes.

In general, the parameters characterizing a -class will form a quadruple (dim,\,system, \mbox{q-class},\,z-class) where dim is the associated dimension, system is the number of the associated crystal system, q-class is the number of the associated -class within the crystal system, and z-class is the number of the -class within the -class. However, in case of dimensions 2 or 3, a -class may also be characterized by a pair (dim, IT-number) where IT-number is the number in the International Tables Hah83 of any space-group type lying in that -class, or just by the Hermann-Mauguin symbol of any space-group type lying in that -class.

DisplayZClass( dim, system, q-class, z-class )
DisplayZClass( dim, IT-number )
DisplayZClass( Hermann-Mauguin-symbol )

DisplayZClass displays for the specified -class essentially the same information as is provided for that -class in Table 1 of BBNWZ78 (except for the generating matrices of a class representative group given there), namely

• for dimensions 2 and 3, the Hermann-Mauguin symbol of a representative space-group type which belongs to that -class,

• the Bravais type,

• some decomposability information,

• the number of space-group types belonging to the -class,

• the size of the associated cohomology group.

For details see BBNWZ78.

    gap> DisplayZClass( 2, 3 );
#I    Z-class (2,2,1,1) = Z(pm): Bravais type II/I; fully Z-reducible;
#I     2 space groups; cohomology group size 2
gap> DisplayZClass( "F-43m" );
#I    Z-class (3,7,4,2) = Z(F-43m): Bravais type VI/II; Z-irreducible;
#I     2 space groups; cohomology group size 2
gap> DisplayZClass( 4, 2, 3, 2 );
#I    Z-class B (4,2,3,2): Bravais type II/II; Z-decomposable;
#I     2 space groups; cohomology group size 4
gap> DisplayZClass( 4, 21, 3, 1 );
#I   *Z-class (4,21,3,1): Bravais type XVI/I; Z-reducible;
#I     1 space group; cohomology group size 1 

MatGroupZClass( dim, system, q-class, z-class )
MatGroupZClass( dim, IT-number )
MatGroupZClass( Hermann-Mauguin-symbol )

MatGroupZClass returns a dim × dim matrix group M, say, which is a representative of the specified -class. Its generators satisfy the defining relators of the finitely presented group which may be computed by calling the FpGroupQClass function (see above) for the -class which contains the given -class.

The generators of M are the same matrices as those given in Table 1 of BBNWZ78. Note, however, that they will be listed in reverse order to keep them in parallel to the abstract generators provided by the FpGroupQClass function (see above).

Besides of the usual components, the group record of M will have an additional component M.crZClass which saves a list of the parameters that specify the given -class. (In fact, in order to make the resulting group record consistent with those returned by the NormalizerZClass or ZClassRepsDadeGroup functions described below, it also will have an additional component M.crConjugator containing just the identity element of M.)

    gap> M := MatGroupZClass( 4, 20, 3, 1 );
MatGroupZClass( 4, 20, 3, 1 )
gap> for g in M.generators do
>  Print( "\n" ); PrintArray( g ); od; Print( "\n" );

[ [   0,   1,   0,   0 ],
[  -1,   0,   0,   0 ],
[   0,   0,  -1,  -1 ],
[   0,   0,   0,   1 ] ]

[ [  -1,   0,   0,   0 ],
[   0,  -1,   0,   0 ],
[   0,   0,  -1,  -1 ],
[   0,   0,   1,   0 ] ]

gap> M.size;
12
gap> M.crZClass;
[ 4, 20, 3, 1 ] 

NormalizerZClass( dim, system, q-class, z-class )
NormalizerZClass( dim, IT-number )
NormalizerZClass( Hermann-Mauguin-symbol )

NormalizerZClass returns the normalizer N, say, in GL(dim,ℤ) of the representative dim × dim matrix group which is constructed by the MatGroupZClass function (see above).

If the size of N is finite, then N again lies in some -class. In this case, the group record of N will contain two additional components N.crZClass and N.crConjugator which provide the parameters of that -class and a matrix g ∈ GL(dim,ℤ), respectively, such that N = g-1 R g, where R is the representative group of that -class.

    gap> N := NormalizerZClass( 4, 20, 3, 1 );
NormalizerZClass( 4, 20, 3, 1 )
gap> for g in N.generators do
>  Print( "\n" ); PrintArray( g ); od; Print( "\n" );

[ [   1,   0,   0,   0 ],
[   0,   1,   0,   0 ],
[   0,   0,   1,   0 ],
[   0,   0,  -1,  -1 ] ]

[ [   1,   0,   0,   0 ],
[   0,  -1,   0,   0 ],
[   0,   0,  -1,  -1 ],
[   0,   0,   1,   0 ] ]

[ [   0,   1,   0,   0 ],
[  -1,   0,   0,   0 ],
[   0,   0,   1,   0 ],
[   0,   0,   0,   1 ] ]

[ [  -1,   0,   0,   0 ],
[   0,  -1,   0,   0 ],
[   0,   0,  -1,   0 ],
[   0,   0,   0,  -1 ] ]

gap> N.size;
96
gap> N.crZClass;
[ 4, 20, 22, 1 ]
gap> N.crConjugator = N.identity;
true 

    gap> L := NormalizerZClass( 3, 42 );
NormalizerZClass( 3, 3, 2, 4 )
gap> L.size;
16
gap> L.crZClass;
[ 3, 4, 7, 2 ]
gap> L.crConjugator;
[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, -1 ] ]
gap> M := NormalizerZClass( "C2/m" );
Group( [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ],
[ [ 0, -1, 0 ], [ -1, 0, 0 ], [ 0, 0, -1 ] ],
[ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 1 ] ],
[ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -1, -1, 1 ] ],
[ [ 0, 1, -1 ], [ 1, 0, -1 ], [ 0, 0, -1 ] ] )
gap> M.size;
"infinity"
gap> IsBound( M.crZClass );
false 

NrSpaceGroupTypesZClass( dim, system, q-class, z-class )
NrSpaceGroupTypesZClass( dim, IT-number )
NrSpaceGroupTypesZClass( Hermann-Mauguin-symbol )

NrSpaceGroupTypes returns the number of space-group types within the given -class. It can be used to formulate loops over the space-group types.

    gap> N := NrSpaceGroupTypesZClass( 4, 4, 1, 1 );
13 

Some of the -classes of dimension d, say, are maximal'' in the sense that the groups in these classes are maximal finite subgroups of GL(d,ℤ). Generalizing a term which is being used for dimension 4, we call the representatives of these maximal -classes the Dade groups'' of dimension d.

NrDadeGroups( dim )

NrDadeGroups returns the number of Dade groups of dimension dim. It can be used to formulate loops over the Dade groups.

There are 2, 4, and 9 Dade groups of dimension 2, 3, and 4, respectively.

    gap> NrDadeGroups( 4 );
9 

DadeGroup( dim, n )

DadeGroup returns the nth Dade group of dimension dim.

    gap> D := DadeGroup( 4, 7 );
MatGroupZClass( 4, 31, 7, 2 ) 

DadeGroupNumbersZClass( dim, system, q-class, z-class )
DadeGroupNumbersZClass( dim, IT-number )
DadeGroupNumbersZClass( Hermann-Mauguin-symbol )

DadeGroupNumbersZClass returns the set of all those integers ni for which the nith Dade group of dimension dim contains a subgroup which, in GL(dim,ℤ), is conjugate to the representative group of the given -class.

    gap> dadeNums := DadeGroupNumbersZClass( 4, 4, 1, 2 );
[ 1, 5, 8 ]
gap> for d in dadeNums do
>     D := DadeGroup( 4, d );
>     Print( D, " of size ", Size( D ), "\n" );
> od;
MatGroupZClass( 4, 20, 22, 1 ) of size 96
MatGroupZClass( 4, 30, 13, 1 ) of size 288
MatGroupZClass( 4, 32, 21, 1 ) of size 384 

ZClassRepsDadeGroup( dim, system, q-class, z-class, n )
ZClassRepsDadeGroup( dim, IT-number, n )
ZClassRepsDadeGroup( Hermann-Mauguin-symbol, n )

ZClassRepsDadeGroup determines in the nth Dade group of dimension dim all those conjugacy classes whose groups are, in GL(dim,ℤ), conjugate to the -class representative group R, say, of the given -class. It returns a list of representative groups of these conjugacy classes.

Let M be any group in the resulting list. Then the group record of M provides two components M.crZClass and M.crConjugator which contain the list of -class parameters of R and a suitable matrix g from GL(dim,ℤ), respectively, such that M equals g-1 R g.

    gap> DadeGroupNumbersZClass( 2, 2, 1, 2 );
[ 1, 2 ]
gap> ZClassRepsDadeGroup( 2, 2, 1, 2, 1 );
[ MatGroupZClass( 2, 2, 1, 2 )^[ [ 0, 1 ], [ -1, 0 ] ] ]
gap> ZClassRepsDadeGroup( 2, 2, 1, 2, 2 );
[ MatGroupZClass( 2, 2, 1, 2 )^[ [ 1, -1 ], [ 0, -1 ] ],
MatGroupZClass( 2, 2, 1, 2 )^[ [ 1, 0 ], [ -1, 1 ] ] ]
gap> R := last;;
gap> R.crZClass;
[ 2, 2, 1, 2 ]
gap> R.crConjugator;
[ [ 1, 0 ], [ -1, 1 ] ] 

The following functions are functions of space-group types.

In general, the parameters characterizing a space-group type will form a quintuple (dim, system,\,q-class,\,z-class,\,sg-type) where dim is the associated dimension, system is the number of the associated crystal system, q-class is the number of the associated -class within the crystal system, z-class is the number of the -class within the -class, and sg-type is the space-group type within the -class. However, in case of dimensions 2 or 3, you may instead specify a -class by a pair (dim, IT-number) or by its Hermann-Mauguin symbol (as described above). Then the function will handle the first space-group type within that -class, i.e., sg-type = 1, that is, the corresponding symmorphic space group (split extension).

DisplaySpaceGroupType( dim, system, q-class, z-class, sg-type )
DisplaySpaceGroupType( dim, IT-number )
DisplaySpaceGroupType( Hermann-Mauguin-symbol )

DisplaySpaceGroupType displays for the specified space-group type some of the information which is provided for that space-group type in Table 1 of BBNWZ78, namely

• the orbit size associated with that space-group type and,

• for dimensions 2 and 3, the IT-number and the Hermann-Mauguin symbol.

For details see BBNWZ78.

    gap> DisplaySpaceGroupType( 2, 17 );
#I     Space-group type (2,4,4,1,1); IT(17) = p6mm; orbit size 1
gap> DisplaySpaceGroupType( "Pm-3" );
#I     Space-group type (3,7,2,1,1); IT(200) = Pm-3; orbit size 1
gap> DisplaySpaceGroupType( 4, 32, 10, 2, 4 );
#I    *Space-group type (4,32,10,2,4); orbit size 18
gap> DisplaySpaceGroupType( 3, 6, 1, 1, 4 );
#I    *Space-group type (3,6,1,1,4); IT(169) = P61, IT(170) = P65;
#I      orbit size 2; fp-free 

DisplaySpaceGroupGenerators( dim, system, q-class, z-class, sg-type )
DisplaySpaceGroupGenerators( dim, IT-number )
DisplaySpaceGroupGenerators( Hermann-Mauguin-symbol )

DisplaySpaceGroupGenerators displays the non-translation generators of a representative space group of the specified space-group type without actually constructing that matrix group.

In more details: Let n = dim be the given dimension, and let M1, ..., Mr be the generators of the representative n × n matrix group of the given -class (this is the group which you will get if you call the MatGroupZClass function (see above) for that -class). Then, for the given space-group type, the SpaceGroup function described below will construct as representative of that space-group type an (n+1) × (n+1) matrix group which is generated by the n translations which are induced by the (standard) basis vectors of the n-dimensional Euclidian space, and r additional matrices S1, ..., Sr of the form Si = [ \catcode|=12 \begintabular{cc} \catcode\=13 M_i & t_i \cr \hline 0 & 1 \endtabular ], where the n × n submatrices Mi are as defined above, and the ti are n-columns with rational entries. The DisplaySpaceGroupGenerators function saves time by not constructing the group, but just displaying the r matrices S1, ..., Sr.

    gap> DisplaySpaceGroupGenerators( "P61" );
#I  The non-translation generators of SpaceGroup( 3, 6, 1, 1, 4 ) are

[ [   -1,    0,    0,    0 ],
[    0,   -1,    0,    0 ],
[    0,    0,    1,  1/2 ],
[    0,    0,    0,    1 ] ]

[ [    0,   -1,    0,    0 ],
[    1,   -1,    0,    0 ],
[    0,    0,    1,  1/3 ],
[    0,    0,    0,    1 ] ] 

SpaceGroup( dim, system, q-class, z-class, sg-type )
SpaceGroup( dim, IT-number )
SpaceGroup( Hermann-Mauguin-symbol )

SpaceGroup returns a (dim+1) × (dim+1 ) matrix group S, say, which is a representative of the given space-group type (see also the description of the DisplaySpaceGroupGenerators function above).

    gap> S := SpaceGroup( "P61" );
SpaceGroup( 3, 6, 1, 1, 4 )
gap> for s in S.generators do
>  Print( "\n" ); PrintArray( s ); od; Print( "\n" );

[ [   -1,    0,    0,    0 ],
[    0,   -1,    0,    0 ],
[    0,    0,    1,  1/2 ],
[    0,    0,    0,    1 ] ]

[ [    0,   -1,    0,    0 ],
[    1,   -1,    0,    0 ],
[    0,    0,    1,  1/3 ],
[    0,    0,    0,    1 ] ]

[ [  1,  0,  0,  1 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  0 ],
[  0,  0,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  1 ],
[  0,  0,  1,  0 ],
[  0,  0,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  1 ],
[  0,  0,  0,  1 ] ]

gap> S.crSpaceGroupType;
[ 3, 6, 1, 1, 4 ] 

Besides of the usual components, the resulting group record of S contains an additional component S.crSpaceGroupType which saves a list of the parameters that specify the given space-group type.

Moreover, it contains, in form of a finitely presented group, a presentation of S which is satisfied by the matrix generators. If the factor group of S by its translation normal subgroup is solvable then this presentation is chosen such that it is a polycyclic power commutator presentation. The proper way to access this presentation is to call the following function.

FpGroup( S )

FpGroup returns a finitely presented group G, say, which is isomorphic to S, where S is expected to be a space group. It is chosen such that there is an isomrphism from G to S which maps each generator of G onto the corresponding generator of S. This means, in particular, that the matrix generators of S satisfy the relators of G.

    gap> G := FpGroup( S );
Group( g1, g2, g3, g4, g5 )
gap> for rel in G.relators do Print( rel, "\n" ); od;
g1^2*g5^-1
g2^3*g5^-1
g2^-1*g1^-1*g2*g1
g3^-1*g1^-1*g3*g1*g3^2
g3^-1*g2^-1*g3*g2*g4*g3^2
g4^-1*g1^-1*g4*g1*g4^2
g4^-1*g2^-1*g4*g2*g4*g3^-1
g4^-1*g3^-1*g4*g3
g5^-1*g1^-1*g5*g1
g5^-1*g2^-1*g5*g2
g5^-1*g3^-1*g5*g3
g5^-1*g4^-1*g5*g4
gap> # Verify that the matrix generators of S satisfy the relators of G.
gap> ForAll( G.relators,
>  rel -> MappedWord( rel, G.generators, S.generators ) = S.identity );
true 

TransposedSpaceGroup( dim, system, q-class, z-class, sg-type )
TransposedSpaceGroup( dim, IT-number )
TransposedSpaceGroup( Hermann-Mauguin-symbol )
TransposedSpaceGroup( S )

TransposedSpaceGroup returns a matrix group T, say, whose generators are just the transposed generators (in the same order) of the corresponding space group S specified by the arguments. As for S, you may get a finite presentation for T via the FpGroup function.

The purpose of this function is explicitly discussed in the introduction to this section.

    gap> T := TransposedSpaceGroup( S );
TransposedSpaceGroup( 3, 6, 1, 1, 4 )
gap> for m in T.generators do
>  Print( "\n" ); PrintArray( m ); od; Print( "\n" );

[ [   -1,    0,    0,    0 ],
[    0,   -1,    0,    0 ],
[    0,    0,    1,    0 ],
[    0,    0,  1/2,    1 ] ]

[ [    0,    1,    0,    0 ],
[   -1,   -1,    0,    0 ],
[    0,    0,    1,    0 ],
[    0,    0,  1/3,    1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  0 ],
[  1,  0,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  0 ],
[  0,  1,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  0 ],
[  0,  0,  1,  1 ] ] 

## 38.14 The Small Groups Library

This library contains all groups of order at most 1000 except for 512 and 768 up to isomorphism. There are a total of 174366 such groups.

SmallGroup( size, i )

The function SmallGroup( size, i ) returns the ith group of order size in the catalogue. It will return an AgGroup, if the group is soluble and a PermGroup otherwise.

NumberSmallGroups( size )

The function NumberSmallGroups( size ) returns the number of groups of the order size.

AllSmallGroups( size )

The function AllSmallGroups( size ) returns the list of all groups of the order size.

UnloadSmallGroups( list of sizes )

It is possible to work with the catalogue of groups of small order just using the functions described above. However, the catalogue is rather large even though the groups are stored in a very compact description. Thus it might be helpful for a space efficient usage of the catalogue, to know a little bit about unloading parts of the catalogue by hand.

At the first call of one of the functions described above, the groups of order size are loaded and stored in a compact description. GAP will not unload them itsself again. Thus if one calls one of the above functions for a lot of different orders, then all the groups of these orders are stored. Even though the description of the groups is space efficient, this might use a lot of space. For example, if one uses the above functions to load the complete catalogue, then GAP will grow to about 12 MB of workspace.

Thus it might be interesting to unload the groups of some orders again, if they are not used anymore. This can be done by calling the function UnloadSmallGroups( list of sizes )

If the groups of order size are unloaded by hand, then GAP will of course load them again at the next call of SmallGroup( size, i ) or one of the other functions described at the beginning of this section.

IdGroup( G )

Let G be a PermGroup or AgGroup of order at most 1000, but not of order 256, 512 or 768. Then the function call IdGroup( G ) returns a tuple [size, i] meaning that G is isomorphic to the i-th group in the catalogue of groups of order size.

Note that this package calls and uses the ANUPQ share library of GAP in a few cases.

gap3-jm
24 Apr 2021