Let $G$ be the Heisenberg group of order $p^{3}$. This is the subgroup

$$G=\left\{\left(\begin{array}{ccc} 1 & a & x \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right) \mid a, b, x \in \mathbf{F}_{p}\right\}$$

of $3 \times 3$ matrices over the finite field $\mathbf{F}_{p}$ ( $p$ prime). Let $H$ be the subgroup of $G$ of such matrices with $a=0$.

(i) Find all one dimensional representations of $G$.

[You may assume without proof that $[G, G]$ is equal to the set of matrices in $G$ with $a=b=0 .]$

(ii) Let $\psi: \mathbf{F}_{p}=\mathbf{Z} / p \mathbf{Z} \longrightarrow \mathbf{C}^{*}$ be a non-trivial one dimensional representation of $\mathbf{F}_{p}$, and define a one dimensional representation $\rho$ of $H$ by

$$\rho\left(\begin{array}{lll} 1 & 0 & x \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right)=\psi(x)$$

Show that $V_{\psi}=\operatorname{Ind}_{H}^{G}(\rho)$ is irreducible.

(iii) List all the irreducible representations of $G$ and explain why your list is complete.