GAP Manual: 51.22. 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 T_0 be the subgroup of T that is maximal with the property that T_0 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 T_0 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).