Let $n$ be a positive integer and $G$ be a finite group of $n\times n$ matrices with integer coefficients, i.e. $G\subset\operatorname{GL}_n(\mathbb{Z})$. It is known that for sufficiently large $n$, the maximum order of such a group is $2^nn!$ by Feit (although relying on an unpublished manuscript of Weisfeiler I believe) and it is attained.
I am interested in the coefficients that appear in the matrices of such a group. Specifically, I would like to know how many **distinct** numbers can appear.

Clearly it cannot be more than $2^nn!n^2$ if all coefficients of all matrices are distinct but I have the feeling that such a bound cannot be reached. By considering the companion matrix of a cyclotomic polynomial, I think it is possible to produce roughly $n$ or maybe even $n^2$ distinct coefficients. On the other hand, a group like the symmetric group can be encoded with $0,1$ matrices and the group that attains the $2^nn!$ upper bound only uses $-1,0,1$.

My questions are:

- is it possible to derive a much better bound on the number of distinct coefficients, for example polynomial in $n$ ?

or

- is there an example family of finite groups whose number of distinct coefficients is exponential in $n$ ?

I have the feeling that representation theory could help but I am not well-versed in the theory of finite groups.