 Solution

The translation subgroup $U$ is normal and $G/U\cong {\mathbb{F}}_p^{\times }$, via the upper-left entry $a$. Thus the coset representation is the regular representation of that cyclic quotient. If $\chi$ runs through its $p-1$ complex one-dimensional characters, inflated to $G$, thenEach occurs once: the regular character is $p-1$ at the quotient identity and zero elsewhere, which is also the sum of all quotient characters by finite character orthogonality.