Solution
= Solution
The commutator <subgroup> equals the centre, so
$$
G/[G,G]\cong\mathbb F_p^2.
$$
Fix $\omega=e^{2\pi i/p}$. Every linear character is
$$
\chi_{a,b}(x,y,z)=\omega^{ax+by},\qquad a,b\in\mathbb F_p,
$$
and these $p^2$ distinct characters exhaust the one-dimensional representations.