This is more of a question specifically about how GAP handles calculations with finitely presented groups rather than about group theory.
I have several finite group presentations that I would like to do computations on using GAP. Suppose I have denoted one as G. For this example, suppose that by entering the command 'Size(G);', GAP takes approximately a minute to determine that G is of a large finite order. My understanding (please correct me if I am wrong) is that GAP determined this through a Todd-Coxeter coset enumeration of the trivial subgroup of G.
Ultimately, I have other computations to perform on G which I have been approaching by creating a coset table of G (w.r.t. a particular subgroup H, although taking H to be the trivial subgroup is fine for my purposes), but the command 'CosetTable(G,H);' takes an absurdly long time to complete (for this example, absurdly long is defined to mean longer than 30 minutes).
Is there any way to extract or directly access the coset table GAP generated for the trivial subgroup while computing 'Size(G);', or is there more going on behind the scenes that would make such a coset table useless? In particular, being able to trace the action of words in the free group on the generators of G make the output of 'CosetTable(G,H);' ideal for my research. Is this why it takes so long for GAP to generate this coset table, as it has to determine how the generators of G act on every coset of H, including when I define H to be the trivial subgroup of G?