= 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 $G$, then
$$
\boxed{\mathbb C[G/U]\cong\bigoplus_{\chi\in\widehat{\mathbb F_p^\times}}\chi.}
$$
Each 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>.
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