Q:

Is there any element of order 30 in $GL _{10}(\mathbb{C})$ with INTEGER coefficients?

I can easily find an element of order 30 in $GL _{10}(\mathbb{C})$ as $\begin{pmatrix} \cos\frac{2π}{30} & \sin\frac{π}{15} & 0 \\ \sin\frac{-π}{15} & \cos\frac{π}{15} & 0  \\  0 & 0 & I_8  \end{pmatrix}$  but I want if there is any element of this order with integer coefficients.

A:

An example: $\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}$ where $A=\begin{bmatrix}0&1\\1&0\end{bmatrix}$, $B=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}$, and $C=\begin{bmatrix}0&0&0&0&1\\1&0&0&0&0\\0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\end{bmatrix}$
The blocks have order $2,3,5$ respectively, so when we combine them...

