I had computed with GAP3r3 the following :
F2 := FreeGroup(2);; x := F2.1;; y := F2.2;; w := x*y^-1*x^-1*y;; G := F2 / [w*x*w^-1*y^-1, l^7];; h1_7 := [[1,1],[0,1]]*Z(7)^0;; h2_7 := [[1,0],[-2,1]]*Z(7)^0;; h3_7 := [[-1,0],[0,-1]]*Z(7)^0;; SL2_7 := Group(h1_7,h2_7,h3_7);; PSL2_7 := FactorGroup(SL2_7,Subgroup(SL2_7,[h1_7^0,h3_7]));; psi_7 := GroupHomomorphismByImages(G,PSL2_7,G.generators,PSL2_7.generators);; K_7 := Kernel(psi_7);; R_7 := AbelianInvariantsSubgroupFpGroupMtc(G,K_7);
it gave me a list with 32 zeros and one 7
The same computation with GAP3r4p4 doesn't give me the same result.
Could you help me to solve this problem ?