Computing with Finite Solvable Groups in the Computer Algebra System GAP

Computing with finite solvable groups has a long history in computational group theory. The principal idea of algorithms in this area is to use the structure of the group; that is, algorithms are based on so-called polycyclic sequences. Special polycyclic sequences are a significant refinement of this fundamental data. We survey their main features and exhibit applications in computing Sylow systems, maximal subgroups, minimal generating sets and numbers of generating sets and tuples. Examples of applications based on the GAP system demonstrate their practicality.