[GAP Forum] Computing image of a homomorphism

[Ravi Kulkarni]

Dear GAP Forum,
  I need to calculate the image of a homomorphism of a matrix group:

gap> m1 := [ [0,1,0],[-1,0,0],[0,0,1] ];;
gap> m2 := [ [0,1,0],[0,0,1],[1,0,0] ];;
gap> O := Group(m1,m2);; # one of the two 3-dim irrep of SymmetricGroup(4) (S_4)

# compute the five conjugacy classes and combine them into one list
# skip some code...
gap> ccall := Concatenation(cc1,cc2,cc3,cc4,cc5);

I want to calculate the matrices corresponding to the two dimensional
irrep of S_4. Call this W. Examining the character table (and some
computation) shows that W is a representation of the quotient of S_4
by the group generated by the elements of the fifth conjugacy class
(which has character 2 in the W representation).

gap> H := Group(cc5);;
gap> IsNormal(O,H);
true

gap> OmodH := FactorGroup(O,H);
Group([ f1, f2^2 ])

gap> hom := NaturalHomomorphismByNormalSubgroup(O,H);
CompositionMapping( [ (1,2,6,5), (1,2,3)(4,6,5) ] ->
[ f1, f2^2 ], <action isomorphism> )

Compute the images of elements of O:

gap> imghom := List([1..Size(O)], i -> Image(hom,ccall[i]));
[ <identity> of ..., f1*f2, f1*f2, f1*f2^2, f1*f2^2, f1, f1, f2^2, f2, f2,
  f2^2, f2, f2^2, f2, f2^2, f1, f1, f1*f2, f1*f2, f1*f2^2, f1*f2^2,
  <identity> of ..., <identity> of ..., <identity> of ... ]

Now, what I would like to have are the 2x2 matrices corresponding to
each of the elements of "imghom". I can guess what f1 and f2 are in
this example, but would be wary of doing so for a larger group.

Is there a systematic way of getting the matrices f1 and f2?

Thanks,
  Ravi