[GAP Forum] strange result of FactorGroupFpGroupByRels

[Hebert Pérez-Rosés]

Dear Forum,

I am working with the group 17 of order 108, from the small group library,
which I henceforth denote by H. I am getting a strange result when I try to
factor H, and I wonder if you could help me find an explanation.

The subgroup I, generated by the last generator, F.5, is a normal subgroup
of order 3, and indeed, the factor group H/I is S_3 x S_3, of order 36.

Now, I my understanding is that the group J, generated by [F.1, F.2, F.3,
F,4] should have index 3 in H, but GAP tells me this index is 1. I am
including the code below for reference.

Ultimately, I presume that H is a semidirect product of S_3 x S_3 and C_3,
and I would like to find the homomorphism  phi: S_3 x S_3 ----> Aut(C_3).
How can I do that?

By the way, I am using GAP 4.4.12.

Best regards,

Hebert Pérez-Rosés,
University of Lleida, Spain

===================================

gap> G:= SmallGroup(108,17);
<pc group of size 108 with 5 generators>

gap> H:= Image(IsomorphismFpGroup(G));
<fp group of size 108 on the generators [ F1, F2, F3, F4, F5 ]>

gap> RelatorsOfFpGroup(H);

[ F1^2, F2^-1*F1^-1*F2*F1, F3^-1*F1^-1*F3*F1, F4^-1*F1^-1*F4*F1*F4^-1,
  F5^-1*F1^-1*F5*F1*F5^-1, F2^2, F3^-1*F2^-1*F3*F2*F3^-1,
F4^-1*F2^-1*F4*F2,
  F5^-1*F2^-1*F5*F2*F5^-1, F3^3, F4^-1*F3^-1*F4*F3*F5^-1,
F5^-1*F3^-1*F5*F3,
  F4^3, F5^-1*F4^-1*F5*F4, F5^3 ]

gap> I:= FactorGroupFpGroupByRels(H,[H.5]);
<fp group on the generators [ F1, F2, F3, F4, F5 ]>

gap> StructureDescription(I);
"S3 x S3"

gap> J:= FactorGroupFpGroupByRels(H,[H.1,H.2,H.3,H.4]);
<fp group on the generators [ F1, F2, F3, F4, F5 ]>

gap> StructureDescription(J);
"1"

# At this point I thought that GAP's answer was due to the fact that the
subgroup generated by [H.1, ..., H.4] was not normal, but when I tried to
verify this conjecture, I got:

gap> S:= Subgroup(H, [H.1,H.2,H.3,H.4]);
Group ([ F1, F2, F3, F4 ])

gap> IsNormal(H, S);
true

gap> Index(H, S);
1

# Where is the problem here? Have I missed something?