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:

gap> MinimalGeneratingSet(G);
[ f1, f3 ]

To get relations, first we construct an isomorphic finitely presented
group:

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 ]

Sincerely yours,
Alexander Konovalov

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