Find elements of all normal subgroups and some information about group in GAP

Question

gap> f := FreeGroup( "a", "b" );;
gap> g := f / [ f.1^4, f.2^4, (f.1*f.2)^2, (f.1^-1*f.2)^2 ];
<fp group on the generators [ a, b ]>
gap> a := g.1;; b := g.2;;
gap> NormalSubgroups( g );
[ Group(<fp, no generators known>), Group(<fp, no generators known>), Group(<fp, no generators known>),
  Group(<fp, no generators known>), Group(<fp, no generators known>), Group(<fp, no generators known>),
  Group(<fp, no generators known>), Group(<fp, no generators known>), Group(<fp, no generators known>),
  Group(<fp, no generators known>), Group(<fp, no generators known>) ]

So i get some normal subgroups, but i dont know how to find thei elements, what i should to do?

Also i need find factor-groups, and find all unic irreducible representation of main group. The last one most important

Answer 1

Subgroups of finitely presented groups normally do not show their generators, but you can get them easily as GeneratorsOfGroup (or you could ask for the Elements of a subgroup). When representing small groups (in your example of order 16) as FpGroups It also can be convenient to force element reduction through SetReducedMultiplication before starting any calculations.

In your example:

gap> SetReducedMultiplication(g);
gap> n:=NormalSubgroups(g);;    
gap> List(n,GeneratorsOfGroup);
[ [  ], [ a^-2 ], [ b^-2 ], [ a^-1*b^2*a^-1 ], [ a^-2, b^-2 ],
  [ b*a, b^-1*a^-1 ], [ b*a^-1, b^-1*a ], [ a, b^-2 ], [ a^-2, b ],
  [ a^-2, b*a^-1, b^-1*a^-1 ], [ a, b ] ]
gap> n;    
[ Group([  ]), Group([ a^-2 ]), Group([ b^-2 ]), Group([ a^-1*b^2*a^-1 ]),
  Group([ a^-2, b^-2 ]), Group([ b*a, b^-1*a^-1 ]), Group([ b*a^-1, b^-1*a ]),
  Group([ a, b^-2 ]), Group([ a^-2, b ]), Group([ a^-2, b*a^-1, b^-1*a^-1 ]),
  Group([ a, b ]) ]
gap> List(n,Size);
[ 1, 2, 2, 2, 4, 4, 4, 8, 8, 8, 16 ]
gap> Elements(n[7]);
[ <identity ...>, <[ [ 2, 1 ] ]|b^-1*a>, <[ [ 1, 1 ] ]|b*a^-1>, b^2*a^2 ]

Finally IrreducibleRepresentations(g); will give you a list of the C-reps. up to equivalence, each given as a GAP homomorphism.