# 51.22 ConsiderTableAutomorphisms

`ConsiderTableAutomorphisms( parafus, tableautomorphisms )`

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).

GAP 3.4.4
April 1997