Finite Presentation in GAP of infinite index subgroup Suppose I have a group $G$ with a finite presentation, and I know that $G$ is polycyclic.  So I input the group into GAP via generators and relators.  Now I want to find a presentation of an infinite index subgroup; I can do this by hand (usually), by doing a Reidemeister-Schreier rewriting, and then simplifying the presentation as best I can.  GAP will, in general, not attempt to find presentations of infinite index subgroups.  But if I tell GAP the group is polycyclic, are there various algorithms out there such that GAP can in fact give me a finite presentation for this infinite index subgroup? 
If that is too much, can GAP do it for certain "nice" subgroups (the commutator, etc.)?
 A: To do what you want, you need a tool that takes a finitely presented group, and, given the information that this group is polycyclic, determines a polycyclic presentation for this group. In general, this is a computationally very difficult problem, although in certain special cases it may be much simpler.
The best approach that I know to this problem in general is to apply a polycyclic quotient algorithm, such as the one described in the paper "A Polycyclic Quotient Algorithm" by Eddie Lo (J. Symbolic Computation (1998) 25, 61–97). He implemented this in C and GAP3, which makes it a tad difficult to use on modern day computers.
The idea then is to take your finite presentation, and compute a maximal PC quotient of it. Since you know your group is polycyclic, this will definitely terminate, but it might take a looong time and a lot of memory (and loong could easily mean more than your life time if things are bad... :/). 
But once you have a pc presentation, you can usually work quite effectively with it, with the help of the polycyclic package, as Jack Schmidt already pointed out.
Anyway, if you still want to try to find a pc presentation for a specific group, I implemented a pc quotient algorithm in GAP 4 last year, which can perhaps solve your problem. It's not yet been published, though I hope to get this finally done sometime soon. In the meantime, feel free to get in touch with my privately and perhaps I can help you in concrete cases.
