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is known that for sufficiently large $n$, the maximum order of such a group is $2^nn!$ by Feit (although relying on an unpublished manuscript of Weisfeiler I believe) and it is attained.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.



On the coefficients that appear in finite groups of matrices with integer entries
It is possible to get a number of distinct coefficients exponential in $n$. Here is an example.


Let
$$B = \begin{bmatrix}
1 & -1 & -1 & -1 & \cdots & x\_1 \\
0 & 1 & 0 & 0 & & x\_2 \\
0 & 0 & 1 & 0 & \cdots & x\_3 \\
0 & 0 & 0 & 1 & & \vdots \\
 & \vdots & & & \ddots & x\_{n-1} \\
0 & 0 & 0 & 0 & 0 & 1 
\end{bmatrix} $$
We have
$$B^{-1} = \begin{bmatrix}
1 & 1 & 1 & 1 & \cdots & -S \\
0 & 1 & 0 & 0 & & -x\_2 \\
0 & 0 & 1 & 0 & \cdots & -x\_3 \\
0 & 0 & 0 & 1 & & \vdots \\
 & \vdots & & & \ddots & -x\_{n-1} \\
0 & 0 & 0 & 0 & 0 & 1 
\end{bmatrix} $$
where $S=x\_1+x\_2+\cdots+x\_{n-1}$.


We consider the conjugate by $B$ of the group of diagonal $\pm 1$ matrices, of size $2^n$.
Let $X$ be a diagonal matrix with $X\_{i,i}=\varepsilon\_i \in \{\pm 1\}$ and suppose $\varepsilon\_n=1$. Then we can compute the upper right entry of $B^{-1}XB$ as
$$-S + \sum\_{i=1}^{n-1} \varepsilon\_i x\_i$$
If we take, for example, $x\_i=2^i$, then these sums are all different for the $2^{n-1}$ possible choices of $X$, so at least $2^{n-1}$ different coefficients appear in the matrices of the group.


