Let $$H= \left\{\left.\begin{bmatrix} a & x \\ 0 & b \end{bmatrix} \right| a,b \in (\mathbb{Z}/3\mathbb{Z})^{\times}, x \in \mathbb{Z}/3\mathbb{Z} \right\} \leq GL_2(\mathbb{Z}/3\mathbb{Z}).$$ Show that $H \cong D_6$, the group of symmetries of a regular hexagon.
I started by showing that they are both of order $12$. I also know that I have to show that there is a homomorpshism and that it is bijective but I'm not sure how to find this.
I'll give you a hint:
$\begin{bmatrix}-1&1\\0&-1\end{bmatrix}$ has order $6$. This will give you the rotations (by taking powers), now pick any of the remaining six, to use as a reflection.