Groups defined by a presentation arise naturally in various areas of mathematics. For example, finitely presented groups play a role in Topology. Another class of examples are the early counterexamples of the famous Burnside problem: many of these have natural presentations, so-called finite $L$-presentations. These have finitely many generators, but not necessarily finitely many relators.

We will present an effective algorithm which takes as input a finitely $L$-presented group G and a positive integer $n$. It decides whether the derived length-$n$ quotient $G/G^{(n)}$ is polycyclic and, if so, then it computes a consistent polycyclic presentation for this quotient, enabling one to effectively study $G/G^{(n)}$. Besides the derived series, the algorithm is very flexible and allows studying many other subgroup series.

Our method uses Gröbner bases for modules of the integral group ring of polycyclic groups, and has been implemented in GAP.