= Solution
Take the abelian <subgroup>
$$
A=\{(x,0,z):x,z\in\mathbb F_p\},
$$
of order $p^2$. For each $c\in\mathbb F_p^\times$, define $\theta_c(x,0,z)=\omega^{cz}$ and form the <induced representation>
$$
\rho_c=\operatorname{Ind}_A^G\theta_c.
$$
It has degree $[G:A]=p$. Conjugation by representatives $(0,y,0)$ gives $p$ distinct characters of $A$ because $c\ne0$. The stabilizer of $\theta_c$ is therefore exactly $A$, and the Mackey irreducibility criterion shows that $\rho_c$ is irreducible. The $p-1$ choices of $c$ have distinct central characters.
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