52 Maps and Parametrized Maps

In this chapter, first the data structure of (parametrized) maps is introduced (see More about Maps and Parametrized Maps). Then a description of several functions which mainly deal with parametrized maps follows; these are

basic operations with paramaps (see CompositionMaps, InverseMap, Indirected, ProjectionMap, Parametrized, ContainedMaps, UpdateMap, CommutativeDiagram, TransferDiagram),

functions which inform about ambiguity with respect to a paramap (see Indeterminateness, PrintAmbiguity),

functions used for the construction of powermaps and subgroup fusions (see Powermap, SubgroupFusions and their subroutines InitPowermap, Congruences, ConsiderKernels, ConsiderSmallerPowermaps, PowermapsAllowedBySymmetrisations, InitFusion, CheckPermChar, CheckFixedPoints, TestConsistencyMaps, ConsiderTableAutomorphisms, FusionsAllowedByRestrictions, OrbitPowermaps, OrbitFusions, RepresentativesPowermaps, RepresentativesFusions, Powmap) and

the function ElementOrdersPowermap.

52.1 More about Maps and Parametrized Maps

Besides the characters, powermaps are an important part of a character table. Often their computation is not easy, and in general they cannot be obtained from the matrix of irreducible characters, so it is useful to store them on the table.

If not only a single table is considered but different tables of groups and subgroups are used, also subgroup fusion maps must be known to get informations about the embedding or simply to induce or restrict characters.

These are examples of class functions which are called maps for short; in GAP3, maps are lists: Characters are maps, the lists of element orders, centralizer orders, classlengths are maps, and for a permutation perm of classes, ListPerm( perm ) is a map.

When maps are constructed, in most cases one only knows that the image of any class is contained in a set of possible images, e.g. that the image of a class under a subgroup fusion is in the set of all classes with the same element order. Using further informations, like centralizer orders, powermaps and the restriction of characters, the sets of possible images can be diminished. In many cases, at the end the images are uniquely determined.

For this, many functions do not only work with maps but with parametrized maps (or short paramaps): These are lists whose entries are either the images themselves (i.e. integers for fusion maps, powermaps, element orders etc. and cyclotomics for characters) or lists of possible images. Thus maps are special paramaps. A paramap paramap can be identified with the set of all maps map with map[i] = paramap[i] or map[i] contained in paramap[i]; we say that map is contained in paramap then.

The composition of two paramaps is defined as the paramap that contains all compositions of elements of the paramaps. For example, the indirection of a character by a parametrized subgroup fusion map is the parametrized character that contains all possible restrictions of that character.

52.2 CompositionMaps

CompositionMaps( paramap2, paramap1 )
CompositionMaps( paramap2, paramap1, class )

For parametrized maps paramap1 and paramap2 where paramap[i] is a bound position or a set of bound positions in paramap2, CompositionMaps( paramap2, paramap1 ) is a parametrized map with image CompositionMaps( paramap2, paramap1, class ) at position class.

If paramap1[ class ] is unique, we have

 CompositionMaps( paramap2, paramap1, class ) = paramap2[ paramap1[ class ] ],
otherwise it is the union of paramap2[i] for i in paramap1[ class ].

    gap> map1:= [ 1, [ 2, 3, 4 ], [ 4, 5 ], 1 ];;
gap> map2:= [ [ 1, 2 ], 2, 2, 3, 3 ];;
gap> CompositionMaps( map2, map1 ); CompositionMaps( map1, map2 );
[ [ 1, 2 ], [ 2, 3 ], 3, [ 1, 2 ] ]
[ [ 1, 2, 3, 4 ], [ 2, 3, 4 ], [ 2, 3, 4 ], [ 4, 5 ], [ 4, 5 ] ]

Note: If you want to get indirections of characters which contain unknowns (see chapter Unknowns) instead of sets of possible values, use Indirected Indirected.

52.3 InverseMap

InverseMap( paramap )

InverseMap( paramap )[i] is the unique preimage or the set of all preimages of i under paramap, if there are any; otherwise it is unbound.

(We have CompositionMaps( paramap, InverseMap( paramap ) ) the identity map.)

    gap> t:= CharTable( "2.A5" );; f:= CharTable( "A5" );;
gap> fus:= GetFusionMap( t, f );    # the factor fusion map
[ 1, 1, 2, 3, 3, 4, 4, 5, 5 ]
gap> inverse:= InverseMap( fus );
[ [ 1, 2 ], 3, [ 4, 5 ], [ 6, 7 ], [ 8, 9 ] ]
gap> CompositionMaps( fus, inverse );
[ 1, 2, 3, 4, 5 ]
gap> t.powermap[2];
[ 1, 1, 2, 4, 4, 8, 8, 6, 6 ]
# transfer a powermap up to the factor group:
gap> CompositionMaps( fus, CompositionMaps( last, inverse ) );
[ 1, 1, 3, 5, 4 ]                   # is equal to f.powermap[2]
# transfer a powermap down to the group:
gap> CompositionMaps( inverse, CompositionMaps( last, fus ) );
[ [ 1, 2 ], [ 1, 2 ], [ 1, 2 ], [ 4, 5 ], [ 4, 5 ], [ 8, 9 ],
[ 8, 9 ], [ 6, 7 ], [ 6, 7 ] ]    # contains t.powermap[2]

52.4 ProjectionMap

ProjectionMap( map )

For each image i under the (necessarily not parametrized) map map, ProjectionMap( map )[i] is the smallest preimage of i.

(We have CompositionMaps( map, ProjectionMap( map ) ) the identity map.)

    gap> ProjectionMap( [1,1,1,2,2,2,3,4,5,5,5,6,6,6,7,7,7] );
[ 1, 4, 7, 8, 9, 12, 15 ]

52.5 Parametrized

Parametrized( list )

returns the parametrized cover of list, i.e. the parametrized map with smallest indeterminateness that contains all maps in list. Parametrized is the inverse function of ContainedMaps in the sense that Parametrized( ContainedMaps( paramap ) ) = paramap.

    gap> Parametrized( [ [ 1, 3, 4, 6, 8, 10, 11, 11, 15, 14 ],
>                    [ 1, 3, 4, 6, 8, 10, 11, 11, 14, 15 ],
>                    [ 1, 3, 4, 7, 8, 10, 12, 12, 15, 14 ],
>                    [ 1, 3, 4, 7, 8, 10, 12, 12, 14, 15 ] ] );
[ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ], [ 14, 15 ],
[ 14, 15 ] ]

52.6 ContainedMaps

ContainedMaps( paramap )

returns the set of all maps contained in the parametrized map paramap. ContainedMaps is the inverse function of Parametrized in the sense that Parametrized( ContainedMaps( paramap ) ) = paramap.

    gap> ContainedMaps( [ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ],
> 14, 15 ] );
[ [ 1, 3, 4, 6, 8, 10, 11, 11, 14, 15 ],
[ 1, 3, 4, 6, 8, 10, 11, 12, 14, 15 ],
[ 1, 3, 4, 6, 8, 10, 12, 11, 14, 15 ],
[ 1, 3, 4, 6, 8, 10, 12, 12, 14, 15 ],
[ 1, 3, 4, 7, 8, 10, 11, 11, 14, 15 ],
[ 1, 3, 4, 7, 8, 10, 11, 12, 14, 15 ],
[ 1, 3, 4, 7, 8, 10, 12, 11, 14, 15 ],
[ 1, 3, 4, 7, 8, 10, 12, 12, 14, 15 ] ]

52.7 UpdateMap

UpdateMap( char, paramap, indirected )

improves the paramap paramap using that indirected is the (possibly parametrized) indirection of the character char by paramap.

    gap> s:= CharTable( "S4(4).2" );; he:= CharTable( "He" );;
gap> fus:= InitFusion( s, he );
[ 1, 2, 2, [ 2, 3 ], 4, 4, [ 7, 8 ], [ 7, 8 ], 9, 9, 9, [ 10, 11 ],
[ 10, 11 ], 18, 18, 25, 25, [ 26, 27 ], [ 26, 27 ], 2, [ 6, 7 ],
[ 6, 7 ], [ 6, 7, 8 ], 10, 10, 17, 17, 18, [ 19, 20 ], [ 19, 20 ] ]
gap> Filtered( s.irreducibles, x -> x[1] = 50 );
[ [ 50, 10, 10, 2, 5, 5, -2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1,
10, 2, 2, 2, 1, 1, 0, 0, 0, -1, -1 ],
[ 50, 10, 10, 2, 5, 5, -2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1,
-10, -2, -2, -2, -1, -1, 0, 0, 0, 1, 1 ] ]
gap> UpdateMap( he.irreducibles[2], fus, last[1] + s.irreducibles[1] );
gap> fus;
[ 1, 2, 2, 3, 4, 4, 8, 7, 9, 9, 9, 10, 10, 18, 18, 25, 25,
[ 26, 27 ], [ 26, 27 ], 2, [ 6, 7 ], [ 6, 7 ], [ 6, 7 ], 10, 10,
17, 17, 18, [ 19, 20 ], [ 19, 20 ] ]

52.8 CommutativeDiagram

CommutativeDiagram( paramap1, paramap2, paramap3, paramap4 )
CommutativeDiagram( paramap1, paramap2, paramap3, paramap4, improvements )

If

 CompositionMaps(paramap2,paramap1 ) = CompositionMaps(paramap4,paramap3 )
shall hold, the consistency is checked and the four maps will be improved according to this condition.

If a record improvements with fields imp1, imp2, imp3, imp4 (all lists) is entered as parameter, only diagrams containing elements of impi as positions in the i-th paramap are considered.

CommutativeDiagram returns false if an inconsistency was found, otherwise a record is returned that contains four lists imp1, ..., imp4, where impi is the list of classes where the i-th paramap was improved.

    gap> map1:= [ [ 1, 2, 3 ], [ 1, 3 ] ];;
gap> map2:= [ [ 1, 2 ], 1, [ 1, 3 ] ];;
gap> map3:= [ [ 2, 3 ], 3 ];; map4:= [ , 1, 2, [ 1, 2 ] ];;
gap> CommutativeDiagram( map1, map2, map3, map4 );
rec(
imp1 := [ 2 ],
imp2 := [ 1 ],
imp3 := [  ],
imp4 := [  ] )
gap> imp:= last;; map1; map2; map3; map4;
[ [ 1, 2, 3 ], 1 ]
[ 2, 1, [ 1, 3 ] ]
[ [ 2, 3 ], 3 ]
[ , 1, 2, [ 1, 2 ] ]
gap> CommutativeDiagram( map1, map2, map3, map4, imp );
rec(
imp1 := [  ],
imp2 := [  ],
imp3 := [  ],
imp4 := [  ] )

52.9 TransferDiagram

TransferDiagram( inside1, between, inside2 )
TransferDiagram( inside1, between, inside2, improvements )

Like in CommutativeDiagram, it is checked that

 CompositionMaps( between, inside1 ) = CompositionMaps( inside2, between )
holds for the paramaps inside1, between and inside2, that means the paramap between occurs twice in each commutative diagram.

Additionally, CheckFixedPoints CheckFixedPoints is called.

If a record improvements with fields impinside1, impbetween and impinside2 is specified, only those diagrams with elements of impinside1 as positions in inside1, elements of impbetween as positions in between or elements of impinside2 as positions in inside2 are considered.

When an inconsistency occurs, the program immediately returns false; else it returns a record with lists impinside1, impbetween and impinside2 of found improvements.

    gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );;
gap> fus:= InitFusion( s, ru );;
gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );;
gap> CheckPermChar( s, ru, fus, permchar );; fus;
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20,
[ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ],
[ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ]
gap> TransferDiagram( s.powermap[2], fus, ru.powermap[2] );
rec(
impinside1 := [  ],
impbetween := [ 12, 23 ],
impinside2 := [  ] )
gap> TransferDiagram( s.powermap[3], fus, ru.powermap[3] );
rec(
impinside1 := [  ],
impbetween := [ 14, 24, 25 ],
impinside2 := [  ] )
gap> TransferDiagram( s.powermap[2], fus, ru.powermap[2], last );
rec(
impinside1 := [  ],
impbetween := [  ],
impinside2 := [  ] )
gap> fus;
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, [ 25, 26 ],
[ 25, 26 ], 5, 5, 6, 8, 14, 13, 19, 19, [ 25, 26 ], [ 25, 26 ], 27,
27 ]

52.10 Indeterminateness

Indeterminateness( paramap )

returns the indeterminateness of paramap, i.e. the number of maps contained in the parametrized map paramap

    gap> m11:= CharTable( "M11" );; m12:= CharTable( "M12" );;
gap> fus:= InitFusion( m11, m12 );
[ 1, [ 2, 3 ], [ 4, 5 ], [ 6, 7 ], 8, [ 9, 10 ], [ 11, 12 ],
[ 11, 12 ], [ 14, 15 ], [ 14, 15 ] ]
gap> Indeterminateness( fus );
256
gap> TestConsistencyMaps( m11.powermap, fus, m12.powermap );; fus;
[ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ], [ 14, 15 ],
[ 14, 15 ] ]
gap> Indeterminateness( fus );
32

52.11 PrintAmbiguity

PrintAmbiguity( list, paramap )

prints for each character in list its position, its indeterminateness with respect to paramap and the list of ambiguous classes

    gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );;
gap> fus:= InitFusion( s, ru );;
gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );;
gap> CheckPermChar( s, ru, fus, permchar );; fus;
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20,
[ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ],
[ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ]
gap> PrintAmbiguity( Sublist( ru.irreducibles, [ 1 .. 8 ] ), fus );
1 1 [  ]
2 16 [ 16, 17, 26, 27 ]
3 16 [ 16, 17, 26, 27 ]
4 32 [ 12, 14, 23, 24, 25 ]
5 4 [ 12, 23 ]
6 1 [  ]
7 32 [ 12, 14, 23, 24, 25 ]
8 1 [  ]
gap> Indeterminateness( fus );
512

52.12 Powermap

Powermap( tbl, prime )
Powermap( tbl, prime, parameters )

returns a list of possibilities for the prime-th powermap of the character table tbl.

The optional record parameters may have the following fields:

chars:

a list of characters which are used for the check of kernels (see ConsiderKernels), the test of congruences (see Congruences) and the test of scalar products of symmetrisations (see PowermapsAllowedBySymmetrisations); the default is tbl.irreducibles

powermap:

a (parametrized) map which is an approximation of the desired map

decompose:

a boolean; if true, the symmetrisations of chars must have all constituents in chars, that will be used in the algorithm; if chars is not bound and tbl.irreducibles is complete, the default value of decompose is true, otherwise false

quick:

a boolean; if true, the subroutines are called with the option "quick"; especially, a unique map will be returned immediately without checking all symmetrisations; the default value is false

parameters:

a record with fields maxamb, minamb and maxlen which control the subroutine PowermapsAllowedBySymmetrisations: It only uses characters with actual indeterminateness up to maxamb, tests decomposability only for characters with actual indeterminateness at least minamb and admits a branch only according to a character if there is one with at most maxlen possible minus-characters.

    # cf. example in InitPowermap
gap> t:= CharTable( "U4(3).4" );;
gap> pow:= Powermap( t, 2 );
[ [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, 6, 14, 9, 1, 1, 2, 2, 3, 4,
5, 6, 8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, 18,
18, 20, 20, 20, 20, 22, 22, 24, 24, 25, 26, 28, 28, 29, 29 ] ]

52.13 SubgroupFusions

SubgroupFusions( subtbl, tbl )
SubgroupFusions( subtbl, tbl, parameters )

returns the list of all subgroup fusion maps from subtbl into tbl.

The optional record parameters may have the following fields:

chars:

a list of characters of tbl which will be restricted to subtbl, (see FusionsAllowedByRestrictions); the default is tbl.irreducibles

subchars:

a list of characters of subtbl which are constituents of the retrictions of chars, the default is subtbl.irreducibles

fusionmap:

a (parametrized) map which is an approximation of the desired map

decompose:

a boolean; if true, the restrictions of chars must have all constituents in subchars, that will be used in the algorithm; if subchars is not bound and subtbl.irreducibles is complete, the default value of decompose is true, otherwise false

permchar:

a permutation character; only those fusions are computed which afford that permutation character (see CheckPermChar)

quick:

a boolean; if true, the subroutines are called with the option "quick"; especially, a unique map will be returned immediately without checking all symmetrisations; the default value is false

parameters:

a record with fields maxamb, minamb and maxlen which control the subroutine FusionsAllowedByRestrictions: It only uses characters with actual indeterminateness up to maxamb, tests decomposability only for characters with actual indeterminateness at least minamb and admits a branch only according to a character if there is one with at most maxlen possible restrictions.

    # cf. example in FusionsAllowedByRestrictions
gap> s:= CharTable( "U3(3)" );; t:= CharTable( "J4" );;
gap> SubgroupFusions( s, t, rec( quick:= true ) );
[ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ],
[ 1, 2, 4, 4, 5, 5, 6, 10, 13, 12, 14, 14, 21, 21 ],
[ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 15, 15, 22, 22 ],
[ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 16, 16, 22, 22 ],
[ 1, 2, 4, 4, 6, 6, 6, 10, 13, 12, 15, 15, 22, 22 ],
[ 1, 2, 4, 4, 6, 6, 6, 10, 13, 12, 16, 16, 22, 22 ] ]

52.14 InitPowermap

InitPowermap( tbl, prime )

computes a (probably parametrized, see More about Maps and Parametrized Maps) first approximation of of the prime-th powermap of the character table tbl, using that for any class i of tbl, the following properties hold:

The centralizer order of the image is a multiple of the centralizer order of i. If the element order of i is relative prime to prime, the centralizer orders of i and its image must be equal.

If prime divides the element order x of the class i, the element order of its image must be <x> / prime; otherwise the element orders of i and its image must be equal. Of course, this is used only if the element orders are stored on the table.

If no prime-th powermap is possible because of these properties, false is returned. Otherwise InitPowermap returns the parametrized map.

    # cf. example in Powermap
gap> t:= CharTable( "U4(3).4" );;
gap> pow:= InitPowermap( t, 2 );
[ 1, 1, 3, 4, 5, [ 2, 16 ], [ 2, 16, 17 ], 8, 3, [ 3, 4 ],
[ 11, 12 ], [ 11, 12 ], [ 6, 7, 18, 19, 30, 31, 32, 33 ], 14,
[ 9, 20 ], 1, 1, 2, 2, 3, [ 3, 4, 5 ], [ 3, 4, 5 ],
[ 6, 7, 18, 19, 30, 31, 32, 33 ], 8, 9, 9, [ 9, 10, 20, 21, 22 ],
[ 11, 12 ], [ 11, 12 ], 16, 16, [ 2, 16 ], [ 2, 16 ], 17, 17,
[ 6, 18, 30, 31, 32, 33 ], [ 6, 18, 30, 31, 32, 33 ],
[ 6, 7, 18, 19, 30, 31, 32, 33 ], [ 6, 7, 18, 19, 30, 31, 32, 33 ],
20, 20, [ 9, 20 ], [ 9, 20 ], [ 9, 10, 20, 21, 22 ],
[ 9, 10, 20, 21, 22 ], 24, 24, [ 15, 25, 26, 40, 41, 42, 43 ],
[ 15, 25, 26, 40, 41, 42, 43 ], [ 28, 29 ], [ 28, 29 ], [ 28, 29 ],
[ 28, 29 ] ]
# continued in Congruences

InitPowermap is used by Powermap Powermap.

52.15 Congruences

Congruences( tbl, chars, prime\_powermap, prime )
Congruences( tbl, chars, prime\_powermap, prime, "quick" )

improves the parametrized map prime\_powermap (see More about Maps and Parametrized Maps) that is an approximation of the prime-th powermap of the character table tbl:

For G a group with character table tbl, g ∈ G and a character χ of tbl, the congruence

 GaloisCyc( \chi(g), prime ) ≡χ(gprime)(mod prime)
holds; if the representative order of g is relative prime to prime, we have
 GaloisCyc( \chi(g), prime ) = \chi(g^{prime}).
Congruences checks these congruences for the (virtual ) characters in the list chars.

If "quick" is specified, only those classes are considered for which prime\_powermap is ambiguous.

If there are classes for which no image is possible, false is returned, otherwise Congruences returns true.

    # see example in InitPowermap
gap> Congruences( t, t.irreducibles, pow, 2 ); pow;
true
[ 1, 1, 3, 4, 5, [ 2, 16 ], [ 2, 16, 17 ], 8, 3, 4, 11, 12,
[ 6, 7, 18, 19 ], 14, [ 9, 20 ], 1, 1, 2, 2, 3, 4, 5,
[ 6, 7, 18, 19 ], 8, 9, 9, [ 10, 21 ], 11, 12, 16, 16, [ 2, 16 ],
[ 2, 16 ], 17, 17, [ 6, 18 ], [ 6, 18 ], [ 6, 7, 18, 19 ],
[ 6, 7, 18, 19 ], 20, 20, [ 9, 20 ], [ 9, 20 ], 22, 22, 24, 24,
[ 15, 25, 26 ], [ 15, 25, 26 ], 28, 28, 29, 29 ]
# continued in ConsiderKernels

Congruences is used by Powermap Powermap.

52.16 ConsiderKernels

ConsiderKernels( tbl, chars, prime\_powermap, prime )
ConsiderKernels( tbl, chars, prime\_powermap, prime, "quick" )

improves the parametrized map prime\_powermap (see More about Maps and Parametrized Maps) that is an approximation of the prime-th powermap of the character table tbl:

For G a group with character table tbl, the kernel of each character in the list chars is a normal subgroup of G, so for every g ∈ Kernel( chi ) we have gprimeKernel( chi ).

Depending on the order of the factor group modulo Kernel( chi ), there are two further properties: If the order is relative prime to prime, for each g ∉ Kernel( chi ) the prime-th power is not contained in Kernel( chi ); if the order is equal to prime, the prime-th powers of all elements lie in Kernel( chi ).

If "quick" is specified, only those classes are considered for which prime\_powermap is ambiguous.

If Kernel( chi ) has an order not dividing tbl.order for an element chi of chars, or if no image is possible for a class, false is returned; otherwise ConsiderKernels returns true.

Note that chars must consist of ordinary characters, since the kernel of a virtual character is not defined.

    # see example in Congruences
gap> ConsiderKernels( t, t.irreducibles, pow, 2 ); pow;
true
[ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, [ 6, 7 ], 14, 9, 1, 1, 2, 2,
3, 4, 5, [ 6, 7, 18, 19 ], 8, 9, 9, [ 10, 21 ], 11, 12, 16, 16,
[ 2, 16 ], [ 2, 16 ], 17, 17, [ 6, 18 ], [ 6, 18 ],
[ 6, 7, 18, 19 ], [ 6, 7, 18, 19 ], 20, 20, [ 9, 20 ], [ 9, 20 ],
22, 22, 24, 24, [ 15, 25, 26 ], [ 15, 25, 26 ], 28, 28, 29, 29 ]
# continued in PowermapsAllowedBySymmetrisations

ConsiderKernels is used by Powermap Powermap.

52.17 ConsiderSmallerPowermaps

ConsiderSmallerPowermaps( tbl, prime\_powermap, prime )
ConsiderSmallerPowermaps( tbl, prime\_powermap, prime, "quick" )

improves the parametrized map prime\_powermap (see chapter Maps and Parametrized Maps) that is an approximation of the prime-th powermap of the character table tbl:

If <prime> > tbl.orders[i] for a class i, try to improve prime\_powermap at class i using that for g in class i, giprime = giprime mod tbl.orders[i] holds;
so if the (prime mod tbl.orders[i])-th powermap at class i is determined by the maps stored in tbl.powermap, this information is used.

If "quick" is specified, only those classes are considered for which prime\_powermap is ambiguous.

If there are classes for which no image is possible, false is returned, otherwise true.

Note: If tbl.orders is unbound, true is returned without tests.

    gap> t:= CharTable( "3.A6" );; init:= InitPowermap( t, 5 );;
gap> Indeterminateness( init );
4096
gap> ConsiderSmallerPowermaps( t, init, 5 );;
gap> Indeterminateness( init );
256 

ConsiderSmallerPowermaps is used by Powermap Powermap.

52.18 InitFusion

InitFusion( subtbl, tbl )

computes a (probably parametrized, see More about Maps and Parametrized Maps) first approximation of of the subgroup fusion from the character table subtbl into the character table tbl, using that for any class i of subtbl, the centralizer order of the image is a multiple of the centralizer order of i and the element order of i is equal to the element order of its image (used only if element orders are stored on the tables).

If no fusion map is possible because of these properties, false is returned. Otherwise InitFusion returns the parametrized map.

    gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );;
gap> fus:= InitFusion( s, ru );
[ 1, 2, 2, 4, [ 5, 6 ], [ 5, 6, 7, 8 ], [ 5, 6, 7, 8 ], [ 9, 10 ],
11, 14, 14, [ 13, 14, 15 ], [ 16, 17 ], [ 18, 19 ], 20, [ 25, 26 ],
[ 25, 26 ], [ 5, 6 ], [ 5, 6 ], [ 5, 6 ], [ 5, 6, 7, 8 ],
[ 13, 14, 15 ], [ 13, 14, 15 ], [ 18, 19 ], [ 18, 19 ], [ 25, 26 ],
[ 25, 26 ], [ 27, 28, 29 ], [ 27, 28, 29 ] ]

InitFusion is used by SubgroupFusions SubgroupFusions.

52.19 CheckPermChar

CheckPermChar( subtbl, tbl, fusionmap, permchar )

tries to improve the parametrized fusion fusionmap (see Chapter Maps and Parametrized Maps) from the character table subtbl into the character table tbl using the permutation character permchar that belongs to the required fusion: A possible image x of class i is excluded if class i is too large, and a possible image y of class i is the right image if y must be the image of all classes where y is a possible image.

CheckPermChar returns true if no inconsistency occurred, and false otherwise.

    gap> fus:= InitFusion( s, ru );; # cf. example in InitFusion
gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );;
gap> CheckPermChar( s, ru, fus, permchar );; fus;
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20,
[ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ],
[ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ]

CheckPermChar is used by SubgroupFusions SubgroupFusions.

52.20 CheckFixedPoints

CheckFixedPoints( inside1, between, inside2 )

If the parametrized map (see More about Maps and Parametrized Maps) between transfers the parametrized map inside1 to inside2, i.e. <inside2> o between = between o inside1, between must map fixed points of inside1 to fixed points of inside2. Using this property, CheckFixedPoints tries to improve between and inside2.

If an inconsistency occurs, false is returned. Otherwise, CheckFixedPoints returns the list of classes where improvements were found.

    gap> s:= CharTable( "L4(3).2_2" );; o7:= CharTable( "O7(3)" );;
gap> fus:= InitFusion( s, o7 );;
gap> CheckFixedPoints( s.powermap[5], fus, o7.powermap[5] );
[ 48, 49 ]
gap> fus:= InitFusion( s, o7 );; Sublist( fus, [ 48, 49 ] );
[ [ 54, 55, 56, 57 ], [ 54, 55, 56, 57 ] ]
gap> CheckFixedPoints( s.powermap[5], fus, o7.powermap[5] );
[ 48, 49 ]
gap> Sublist( fus, [ 48, 49 ] );
[ [ 56, 57 ], [ 56, 57 ] ]

CheckFixedPoints is used by SubgroupFusions SubgroupFusions.

52.21 TestConsistencyMaps

TestConsistencyMaps( powmap1, fusmap, powmap2 )
TestConsistencyMaps( powmap1, fusmap, powmap2, fus\_imp )

Like in TransferDiagram, it is checked that parametrized maps (see chapter Maps and Parametrized Maps) commute:

For all positions i where both powmap1[i] and powmap2[i] are bound,

 CompositionMaps( fusmap, powmap1[i] ) = CompositionMaps( powmap2[i], fusmap )
shall hold, so fusmap occurs in diagrams for all considered elements of powmap1 resp. powmap2, and it occurs twice in each diagram.

If a set fus\_imp is specified, only those diagrams with elements of fus\_imp as preimages of fusmap are considered.

TestConsistencyMaps stores all found improvements in fusmap and elements of powmap1 and powmap2. When an inconsistency occurs, the program immediately returns false; otherwise true is returned.

TestConsistencyMaps stops if no more improvements of fusmap are possible. E.g. if fusmap was unique from the beginning, the powermaps will not be improved. To transfer powermaps by fusions, use TransferDiagram TransferDiagram.

    gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );;
gap> fus:= InitFusion( s, ru );;
gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );;
gap> CheckPermChar( s, ru, fus, permchar );; fus;
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20,
[ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ],
[ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ]
gap> TestConsistencyMaps( s.powermap, fus, ru.powermap );
true
gap> fus;
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, [ 25, 26 ],
[ 25, 26 ], 5, 5, 6, 8, 14, 13, 19, 19, [ 25, 26 ], [ 25, 26 ], 27,
27 ]
gap> Indeterminateness( fus );
16

TestConsistencyMaps is used by SubgroupFusions SubgroupFusions.

52.22 ConsiderTableAutomorphisms

ConsiderTableAutomorphisms( parafus, tableautomorphisms )

improves the parametrized subgroup fusion map parafus (see More about Maps and Parametrized Maps): Let T be the permutation group that has the list tableautomorphisms as generators, let T0 be the subgroup of T that is maximal with the property that T0 operates on the set of fusions contained in parafus by permutation of images.

ConsiderTableAutomorphisms replaces orbits by representatives at suitable positions so that afterwards exactly one representative of fusion maps (that is contained in parafus) in every orbit under the operation of T0 is contained in parafus.

The list of positions where improvements were found is returned.

    gap> fus:= InitFusion( s, ru );;
gap> permchar:= Sum( Sublist( ru.irreducibles, [ 1, 5, 6 ] ) );;
gap> CheckPermChar( s, ru, fus, permchar );; fus;
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20,
[ 25, 26 ], [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ],
[ 18, 19 ], [ 25, 26 ], [ 25, 26 ], 27, 27 ]
gap> ConsiderTableAutomorphisms( fus, ru.automorphisms );
[ 16 ]
gap> fus;
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, [ 13, 15 ], 16, [ 18, 19 ], 20,
25, [ 25, 26 ], 5, 5, 6, 8, 14, [ 13, 15 ], [ 18, 19 ], [ 18, 19 ],
[ 25, 26 ], [ 25, 26 ], 27, 27 ]

ConsiderTableAutomorphisms is used by SubgroupFusions (see SubgroupFusions). Note that the function SubgroupFusions forms orbits of fusion maps under table automorphisms, but it returns all possible fusions. If you want to get only orbit representatives, use the function RepresentativesFusions (see RepresentativesFusions).

52.23 PowermapsAllowedBySymmetrisations

PowermapsAllowedBySymmetrisations( tbl, subchars, chars, pow,
 prime, parameters )

returns a list of (possibly parametrized, see More about Maps and Parametrized Maps) maps map which are contained in the parametrized map pow and which have the property that for all χ in the list chars of characters of the character table tbl, the symmetrizations

 χp- = MinusCharacter( χ, map, prime )
(see MinusCharacter) have nonnegative integral scalar products with all characters in the list subchars.

parameters must be a record with fields

maxlen:

an integer that controls the position where branches take place

contained:

a function, usually ContainedCharacters or ContainedPossibleCharacters; for a symmetrization minus, it returns the list contained( tbl, subchars, minus )

minamb, maxamb:

two arbitrary objects; contained is called only for symmetrizations minus with
 minamb < Indeterminateness( minus ) < maxamb

quick:

a boolean; if it is true, the scalar products of uniquely determined symmetrizations are not checked.

pow will be improved, i.e. is changed by the algorithm.

If there is no character left which allows an immediate improvement but there are characters in chars with indeterminateness of the symmetrizations bigger than parameters.minamb, a branch is necessary. Two kinds of branches may occur: If parameters.contained( tbl, subchars, minus ) has length at most parameters.maxlen, the union of maps allowed by the characters in minus is computed; otherwise a suitable class c is taken which is significant for some character, and the union of all admissible maps with image x on c is computed, where x runs over pow[c].

    # see example in ConsiderKernels
gap> t := CharTable( "U4(3).4" );;
gap> PowermapsAllowedBySymmetrisations(t,t.irreducibles,t.irreducibles,
>      pow, 2, rec( maxlen:=10, contained:=ContainedPossibleCharacters,
>      minamb:= 2, maxamb:= "infinity", quick:= false ) );
[ [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, 6, 14, 9, 1, 1, 2, 2, 3, 4,
5, 6, 8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, 18,
18, 20, 20, 20, 20, 22, 22, 24, 24, 25, 26, 28, 28, 29, 29 ] ]
gap> t.powermap[2] = last[1];
true

52.24 FusionsAllowedByRestrictions

FusionsAllowedByRestrictions( subtbl, tbl, subchars, chars,
 fus, parameters )

returns a list of (possibly parametrized, see More about Maps and Parametrized Maps) maps map which are contained in the parametrized map fus and which have the property that for all χ in the list chars of characters of the character table tbl, the restrictions

 χsubtbl = CompositionMaps( χ, fus )
(see CompositionMaps) have nonnegative integral scalar products with all characters in the list subchars.

parameters must be a record with fields

maxlen:

an integer that controls the position where branches take place

contained:

a function, usually ContainedCharacters or ContainedPossibleCharacters; for a restriction rest, it returns the list contained( subtbl, subchars, rest );

minamb, maxamb:

two arbitrary objects; contained is called only for restrictions rest with minamb < Indeterminateness( rest ) < maxamb;

quick:

a boolean value; if it is true, the scalar products of uniquely determined restrictions are not checked.

fus will be improved, i.e. is changed by the algorithm.

If there is no character left which allows an immediate improvement but there are characters in chars with indeterminateness of the restrictions bigger than parameters.minamb, a branch is necessary. Two kinds of branches may occur: If parameters.contained( tbl, subchars, rest ) has length at most parameters.maxlen, the union of maps allowed by the characters in rest is computed; otherwise a suitable class c is taken which is significant for some character, and the union of all admissible maps with image x on c is computed, where x runs over fus[c].

    gap> s:= CharTable( "U3(3)" );; t:= CharTable( "J4" );;
gap> fus:= InitFusion( s, t );;
gap> TestConsistencyMaps( s.powermap, fus, t.powermap );;
gap> ConsiderTableAutomorphisms( fus, t.automorphisms );; fus;
[ 1, 2, 4, 4, [ 5, 6 ], [ 5, 6 ], [ 5, 6 ], 10, 12, [ 12, 13 ],
[ 14, 15, 16 ], [ 14, 15, 16 ], [ 21, 22 ], [ 21, 22 ] ]
gap> FusionsAllowedByRestrictions( s, t, s.irreducibles,
>             t.irreducibles, fus, rec( maxlen:= 10,
>             contained:= ContainedPossibleCharacters,
>             minamb:= 2, maxamb:= "infinity", quick:= false ) );
[ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ],
[ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 15, 15, 22, 22 ],
[ 1, 2, 4, 4, 6, 6, 6, 10, 12, 13, 16, 16, 22, 22 ] ]
# cf. example in SubgroupFusions

FusionsAllowedByRestrictions is used by SubgroupFusions SubgroupFusions.

52.25 OrbitFusions

OrbitFusions( subtblautomorphisms, fusionmap, tblautomorphisms )

returns the orbit of the subgroup fusion map fusionmap under the operations of maximal admissible subgroups of the table automorphism groups of the character tables. subtblautomorphisms is a list of generators of the automorphisms of the subgroup table, tblautomorphisms is a list of generators of the automorphisms of the supergroup table.

    gap> s:= CharTable( "U3(3)" );; t:= CharTable( "J4" );;
gap> GetFusionMap( s, t );
[ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ]
gap> OrbitFusions( s.automorphisms, last, t.automorphisms );
[ [ 1, 2, 4, 4, 5, 5, 6, 10, 12, 13, 14, 14, 21, 21 ],
[ 1, 2, 4, 4, 5, 5, 6, 10, 13, 12, 14, 14, 21, 21 ] ]

52.26 OrbitPowermaps

OrbitPowermaps( powermap, matautomorphisms )

returns the orbit of the powermap powermap under the operation of the subgroup matautomorphisms of the maximal admissible subgroup of the matrix automorphisms of the corresponding character table.

    gap> t:= CharTable( "3.McL" );;
gap> maut:= MatAutomorphisms( t.irreducibles, [], Group( () ) );
Group( (55,58)(56,59)(57,60)(61,64)(62,65)(63,66), (35,36), (26,29)
(27,30)(28,31)(49,52)(50,53)(51,54), (40,43)(41,44)(42,45), ( 2, 3)
( 5, 6)( 8, 9)(12,13)(15,16)(18,19)(21,22)(24,25)(27,28)(30,31)(33,34)
(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)
(65,66) )
gap> OrbitPowermaps( t.powermap[3], maut );
[ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ],
[ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]

52.27 RepresentativesFusions

RepresentativesFusions( subtblautomorphisms, listoffusionmaps, tblautomorphisms )
RepresentativesFusions( subtbl, listoffusionmaps, tbl )

returns a list of representatives of the list listoffusionmaps of subgroup fusion maps under the operations of maximal admissible subgroups of the table automorphism groups of the character tables. subtblautomorphisms is a list of generators of the automorphisms of the subgroup table, tblautomorphisms is a list of generators of the automorphisms of the supergroup table. if the parameters subtbl and tbl (character tables) are used, the values of subtbl.automorphisms and subtbl.automorphisms will be taken.

    gap> s:= CharTable( "2F4(2)" );; ru:= CharTable( "Ru" );;
gap> SubgroupFusions( s, ru );
[ [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, 25, 26, 5, 5,
6, 8, 14, 13, 19, 19, 26, 25, 27, 27 ],
[ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, 26, 25, 5, 5,
6, 8, 14, 13, 19, 19, 25, 26, 27, 27 ] ]
gap> RepresentativesFusions( s, last, ru );
[ [ 1, 2, 2, 4, 5, 7, 8, 9, 11, 14, 14, 15, 16, 18, 20, 25, 26, 5, 5,
6, 8, 14, 13, 19, 19, 26, 25, 27, 27 ] ]

52.28 RepresentativesPowermaps

RepresentativesPowermaps( listofpowermaps, matautomorphisms )

returns a list of representatives of the list listofpowermaps of powermaps under the operation of a subgroup matautomorphisms of the maximal admissible subgroup of matrix automorphisms of irreducible characters of the corresponding character table.

    gap> t:= CharTable( "3.McL" );;
gap> maut:= MatAutomorphisms( t.irreducibles, [], Group( () ) );
Group( (55,58)(56,59)(57,60)(61,64)(62,65)(63,66), (35,36), (26,29)
(27,30)(28,31)(49,52)(50,53)(51,54), (40,43)(41,44)(42,45), ( 2, 3)
( 5, 6)( 8, 9)(12,13)(15,16)(18,19)(21,22)(24,25)(27,28)(30,31)(33,34)
(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)
(65,66) )
gap> Powermap( t, 3 );
[ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ],
[ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]
gap> RepresentativesPowermaps( last, maut );
[ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]

52.29 Indirected

Indirected( char, paramap )

We have

 Indirected( char, paramap )[i] = char[ paramap[i] ],
if this value is unique; otherwise it is set unknown (see chapter Unknowns). (For a parametrized indirection, see CompositionMaps.)

    gap> m12:= CharTable( "M12" );;
gap> fus:= [ 1, 3, 4, [ 6, 7 ], 8, 10, [ 11, 12 ], [ 11, 12 ],
>            [ 14, 15 ], [ 14, 15 ] ];;  # parametrized subgroup fusion
# from M11
gap> chars:= Sublist( m12.irreducibles, [ 1 .. 6 ] );;
gap> List( chars, x -> Indirected( x, fus ) );
[ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 11, 3, 2, Unknown(1), 1, 0, Unknown(2), Unknown(3), 0, 0 ],
[ 11, 3, 2, Unknown(4), 1, 0, Unknown(5), Unknown(6), 0, 0 ],
[ 16, 0, -2, 0, 1, 0, 0, 0, Unknown(7), Unknown(8) ],
[ 16, 0, -2, 0, 1, 0, 0, 0, Unknown(9), Unknown(10) ],
[ 45, -3, 0, 1, 0, 0, -1, -1, 1, 1 ] ]

52.30 Powmap

Powmap( powermap, n )
Powmap( powermap, n, class )

The first form returns the n-th powermap where powermap is the powermap of a character table (see Character Table Records). If the n-th position in powermap is bound, this map is returned, otherwise it is computed from the (necessarily stored) powermaps of the prime divisors of n.

The second form returns the image of class under the n-th powermap; for any valid class class, we have Powmap( powermap, n )[ class ] = Powmap( powermap, n, class ).

The entries of powermap may be parametrized maps (see More about Maps and Parametrized Maps).

    gap> t:= CharTable( "3.McL" );;
gap> Powmap( t.powermap, 3 );
[ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37, 37,
37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14,
14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ]
gap> Powmap( t.powermap, 27 );
[ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 1, 1, 37, 37,
37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49, 49, 14,
14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ]
gap> Lcm( t.orders ); Powmap( t.powermap, last );
27720
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

52.31 ElementOrdersPowermap

ElementOrdersPowermap( powermap )

returns the list of element orders given by the maps in the powermap powermap. The entries at positions where the powermaps do not uniquely determine the element order are set to unknowns (see chapter Unknowns).

    gap> t:= CharTable( "3.J3.2" );; t.powermap;
[ , [ 1, 2, 1, 2, 5, 6, 7, 3, 4, 10, 11, 12, 5, 6, 8, 9, 18, 19, 17,
10, 11, 12, 13, 14, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 1,
3, 7, 8, 8, 13, 18, 19, 17, 23, 23, 28, 30 ],
[ 1, 1, 3, 3, 1, 1, 1, 8, 8, 10, 10, 10, 3, 3, 15, 15, 7, 7, 7, 20,
20, 20, 8, 8, 10, 10, 10, 30, 30, 28, 28, 32, 32, 32, 35, 36,
35, 38, 39, 36, 37, 37, 37, 38, 38, 47, 46 ],,
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 2, 13, 14, 15, 16, 19, 17, 18,
3, 4, 4, 23, 24, 5, 6, 6, 30, 31, 28, 29, 32, 34, 33, 35, 36,
37, 38, 39, 40, 43, 41, 42, 44, 45, 47, 46 ],,,,,,,,,,,,
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 1, 2, 1, 2, 32, 34, 33, 35,
36, 37, 38, 39, 40, 41, 42, 43, 45, 44, 35, 35 ],,
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 2, 2,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47 ] ]
gap> ElementOrdersPowermap( last );
[ 1, 3, 2, 6, 3, 3, 3, 4, 12, 5, 15, 15, 6, 6, 8, 24, 9, 9, 9, 10,
30, 30, 12, 12, 15, 15, 15, 17, 51, 17, 51, 19, 57, 57, 2, 4, 6, 8,
8, 12, 18, 18, 18, 24, 24, 34, 34 ]
gap> Unbind( t.powermap[17] ); ElementOrdersPowermap( t.powermap );
[ 1, 3, 2, 6, 3, 3, 3, 4, 12, 5, 15, 15, 6, 6, 8, 24, 9, 9, 9, 10,
30, 30, 12, 12, 15, 15, 15, Unknown(11), Unknown(12), Unknown(13),
Unknown(14), 19, 57, 57, 2, 4, 6, 8, 8, 12, 18, 18, 18, 24, 24,
Unknown(15), Unknown(16) ]

gap3-jm
24 Apr 2021