Abstract Algebra Adventures Part 6: The n-cubic groups

When it comes to Group Theory, this series has so far focused on cyclic groups. We've talked much about the orthoplexic groups and the simplexic groups, both of which are cyclic.

But we cannot talk of two infinite polytope families (the orthoplexes and the simplexes) without talking about the third infinite polytope family: the n-cubes. It's just that, well, in the context of polytopic algebras, n-cubes are a bit more special, and it's taken me much longer to figure out how they fit in our puzzle.

So, how do we go about constructing groups whose convex hulls are n-cubes? Do such groups even exist in \(\mathbb{X}_n\)?