First of all really sorry for not a structured question.
I am reading some representations of Generalized Quaternion Group, specifically $Q_{16}$ whose order is $16$. I saw this question where $Q_{16}$ can be realised as matrices over $\mathbb C$. I am wondering if that can be done for integers or in some extension of integers. Specifically my question is
Are there matrices in $GL(n,\mathbb Z)$ or $GL(n,\mathbb Z[\ast])$ for some integer $n$ and some root of unity $\ast$ such that the mtrices generate a group isomorphic to $Q_{16}$?
Sorry again if there is a mistake anywhere. Thanks in advance.
Let $G = \langle a,b \mid a^{2^n}=1,b^2=a^{2^{n-1}},b^{-1}ab=a^{-1} \rangle = Q_{2^{n+1}}$, and let $\omega$ be a primitive $2^n$th root of $1$.
Then $a \mapsto \left(\begin{array}{cc}\omega&0\\0&\omega^{-1}\end{array}\right)$, $b \mapsto \left(\begin{array}{cc}0&1\\-1&0\end{array}\right)$, defines a faithful representation of $G$ over ${\mathbb{Z}}[\omega]$.
As has been pointed out in comments, by increasing $n$ you can get a representation over ${\mathbb Z}$.