Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 19H / d / Solution

With class representatives $(0,0,z)$ and $(x,y,0)$ for $(x,y)\ne (0,0)$, the complete character table is described byfor $a,b\in {\mathbb{F}}_p$, andfor $c\in {\mathbb{F}}_p^{\times }$. There are $p^2+p-1$ rows, matching the number of conjugacy classes, andso no irreducible characters are missing.