Let $H$ be a group with three generators $c, g, h$ and relations $c^{p}=g^{p}=h^{p}=1$, $c g=g c, c h=h c$ and $g h=c h g$ where $p$ is a prime number.

(a) Show that $|H|=p^{3}$. Show that the conjugacy classes of $H$ are the singletons $\{1\},\{c\}, \ldots,\left\{c^{p-1}\right\}$ and the sets $\left\{g^{m} h^{n}, c g^{m} h^{n}, \ldots, c^{p-1} g^{m} h^{n}\right\}$, as $m, n$ range from 0 to $p-1$, but $(m, n) \neq(0,0)$.

(b) Find $p^{2}$ 1-dimensional representations of $H$.

(c) Let $\omega \neq 1$ be a $p$ th root of unity. Show that the following defines an irreducible representation of $H$ on $\mathbb{C}^{p}$ :

$$\begin{gathered} \rho(c)=\omega \mathrm{Id}, \\ \rho(g) \mathbf{e}_{k}=\omega^{k} \mathbf{e}_{k}, \\ \rho(h) \mathbf{e}_{p}=\mathbf{e}_{1} \text { and } \rho(h) \mathbf{e}_{k}=\mathbf{e}_{k+1} \text { if } k<p \end{gathered}$$

where the $\mathbf{e}_{k}$ are the standard basis vectors of $\mathbb{C}^{p}$.

(d) Show that (b) and (c) cover all irreducible isomorphism classes.