Dear Roger Beresford,
let me try to answer to your first question.
(1) How can I extract the defining relationships for a group from GAP? =
Using gmmnn as shorthand for SmallGroup(mm,nn), I particularly need them =
for g2003, g2704, fifteen of the 32-element groups, g3603, & g3609.
For example, let us take the first group you need:
gap> G:=SmallGroup(20,03); <pc group of size 20 with 3 generators>
BTW, with SmallGroup you obtain a group which is given by
power-commutator presentation, so its generating system is not
necessary the minimal one:
[ f1, f3 ]
To get relations, first we construct an isomorphic finitely presented
gap> f:=IsomorphismFpGroup(G); [ f1, f2, f3 ] -> [ F1, F2, F3 ] gap> H:=Image(f); <fp group of size 20 on the generators [ F1, F2, F3 ]>
And now you could see relations:
gap> RelatorsOfFpGroup(H); [ F1^2*F2^-1, F2^-1*F1^-1*F2*F1, F3^-1*F1^-1*F3*F1*F3^-1, F2^2, F3^-1*F2^-1*F3*F2*F3^-3, F3^5 ]