Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent the quotients $G/H_i$ in some computer algebra system like GAP or Magma.

I already tried to do this by defining $G$ as a finitely-presented group in Magma and using its "LowIndexNormalSubgroup"-method. However, the index becomes too large for Magma to handle.

A different approach I came up with is using a faithful representation of $G$ in some $PSL_m(\mathbb{Z}[ \xi_1,...,\xi_n ])$ and then factoring out a family of suitable family of ideals of $\mathbb{Z}[ \xi_1,...,\xi_n ]$. This has been done for 2D in

Mennicke J. Eine Bemerkung über Fuchssche Gruppen. Inventiones mathematicae. 1967 Aug 1;2(4):301-5.

Given that I am far from being an expert in this subject I would like to know whether someone here could help by pointing me in the right direction.