Solution
= Solution
With class representatives $(0,0,z)$ and $(x,y,0)$ for $(x,y)\ne(0,0)$, the complete character table is described by
$$
\chi_{a,b}(0,0,z)=1,\qquad
\chi_{a,b}(x,y,0)=\omega^{ax+by},
$$
for $a,b\in\mathbb F_p$, and
$$
\chi_c(0,0,z)=p\omega^{cz},\qquad
\chi_c(x,y,0)=0
$$
for $c\in\mathbb F_p^\times$. There are $p^2+p-1$ rows, matching the number of conjugacy classes, and
$$
p^2\cdot1^2+(p-1)p^2=p^3=|G|,
$$
so no irreducible characters are missing.