What does "pc group" mean in GAP?

Question

For example, when I enter G:=CyclicGroup(2); in GAP, the program returns with the information <pc group of size 2 with 1 generators>.

Question: What does "pc group" mean in the above?

Answer 1

It stands for "polycyclic group (presentation)".

This question may also be of interest.

If you want to inspect the underlying presentation of the group, one way to do this is as follows:

gap> G:=CyclicGroup(2);
<pc group of size 2 with 1 generators>
gap> H:=Image(IsomorphismFpGroup(G));
<fp group of size 2 on the generators [ F1 ]>
gap> RelatorsOfFpGroup(H);
[ F1^2 ]

This presentation does not necessarily have a minimal possible number of generators - its generators should satisfy different conditions, what makes algorithms to work with such groups very efficient. For example,

gap> G:=DihedralGroup(8);
<pc group of size 8 with 3 generators>
gap> H:=Image(IsomorphismFpGroup(G));
<fp group of size 8 on the generators [ F1, F2, F3 ]>
gap> RelatorsOfFpGroup(H);
[ F1^2, F2^-1*F1^-1*F2*F1*F3^-1, F3^-1*F1^-1*F3*F1, F2^2*F3^-1, 
  F3^-1*F2^-1*F3*F2, F3^2 ]

In some cases, SimplifiedFpGroup can help to get a more human-readable presentation:

gap> S:=SimplifiedFpGroup(H);
<fp group on the generators [ F1, F2 ]>
gap> RelatorsOfFpGroup(S);
[ F1^2, F2^4, (F1*F2)^2 ]

and this is indeed what one would expect to see for $D_8$.