Embedding a finite group into $\operatorname{GL}_n( \mathbb{Z})$

Question

Let $G$ be a finite group. Having a concrete realisation of this finite group as a group of matrices is often helpful for calculations, which motivates the following questions.

Question 1: Is there a good way to obtain the minimal $n$ such that $G$ embedds into $\operatorname{GL}_n(\mathbb{Z})$ (meaning there is an injective homomorphism of G into $\operatorname{GL}_n(\mathbb{Z})$)? Is it possible to obtain such a minimal embedding via GAP?

Question 2: Is there a good way to obtain the minimal $n$ such that $G$ embedds into $\operatorname{GL}_n(\mathbb{Z})$ such that all matrices in the image have only entries 0 or 1 (or entries 0, 1 and -1)? Is it possible to obtain such a minimal embedding via GAP?

Question 3: Is there a canonical such minimal embedding if we impose some further restrictions or nice properties of such maps?

Answer 1

I have no interest or experience with GAP, but here are some partial answers:

1. Every finite subgroup of $GL_n(\mathbb{Q})$ can be conjugated into $GL_n(\mathbb{Z})$. So it is enough to find the minimal dimensional rational representation.

2. Lemma: An 0-1 matrix with a 0-1 inverse is a permutation matrix. So this is asking for finding a minimal faithful action of $G$ on a finite set.

3. Ignoring the faithfulness issue, finding a minimal non-trivial $G$-set is the same as finding a maximal order subgroup of $G$.