F. Harou writes:

I can reformulate more precisely my problem :

first, I have an exact sequence

0 -> K -> G -> H -> 0

H is PSL(2,Z/7Z) so order(H) is 168.
G is < x,y | x^-1 * y * x * y^-1 * x * y * x^-1 * y^-1 * x >

I try to find a presentation of K for computing the rank of K/[K,K].
I have a representation of G in PSL(2,O_3) and the homomorphism
G -> H for this presentation but it seem to give no more information because
the order of G is infinite. What could I do ?

The map you gave yesterday was onto SL(2,7), not PSL(2,7), and with the
given generators it does not appear to be a homomorphic image of G.

gap>         un:=Z(7)^0;
gap>         zero:=0*Z(7);
gap> gi1:=[[un, un], [zero, un]];
gap>          gi2:=[[un, zero], [2*un, un]];
gap> x:=gi1;; y:=gi2;;
gap> x^-1 * y * x * y^-1 * x * y * x^-1 * y^-1 * x;
[ [ Z(7)^3, Z(7)^2 ], [ Z(7)^5, Z(7) ] ]

The value of the relator is not the identity in the image group.
If you tell me the correct images of x and y in G in PSL(2,7), then I can
show you how to calculate a presentation of the kernel K, and hence K/[K,K].

Roughly, you work out the map of G onto the regular permutation of PSL(2,7),
and use that to form a coset table for G, from which you can get a
presentation of the subgroup (which is the stabiliser of a point in the
regular permutation representation - i.e. K).

Derek Holt.