 Solution

There are the $p-1$ inflated quotient characters of dimension one and the single representation $\rho$ of dimension $p-1$ from part (ii). These are distinct: for $p>2$ their dimensions distinguish $\rho$, and for $p=2$ its nontrivial value on $U$ distinguishes it from the trivial quotient character. Their dimension squares sum toThe sum-of-squares theorem for irreducible representations leaves no room for any further irreducible. This includes $p=2$, when $G\cong C_2$ and both representations have dimension one.