I was wondering if anyone here knows of an example of a group $M \leq \mathrm{GL}_n(\mathbb{Z})$ which is

- nilpotent,
- infinite,
- finitely generated,
- virtually abelian,
- irreducible (over $\mathbb{Z}$ or $\mathbb{C}$, shouldn't matter).

It is not hard to find examples that satisfy all but the last property (e.g. suitable affine groups). But so far we could not write down one with all five.

Motivation: A student of mine is developing an algorithm which computes for a given polycyclic group whether it is residually nilpotent. At this point, his algorithm can deal with all examples we have and any that colleagues provided. Yet it definitely has limits, but we have trouble finding an actual concrete example where the algorithm in its current form is not applicable. A group $M$ as described above could be used to construct a kind of "minimal counterexample" where his current algorithm does not work (and then he could either improve the algorithm, or at least discuss the obstacles in a more concrete setting).