  In this paper we study the determinant of irreducible representations of the
generalized symmetric groups $\mathbb{Z}_r \wr S_n$. We give an explicit
formula to compute the determinant of an irreducible representation of
$\mathbb{Z}_r \wr S_n$. Recently, several authors have characterized and
counted the number of irreducible representations of a given finite group with
nontrivial determinant. Motivated by these results, for given integer $n$, $r$
an odd prime and $\zeta$ a nontrivial multiplicative character of $\mathbb{Z}_r
\wr S_n$ with $n<r$, we obtain an explicit formula to compute $N_{\zeta}(n)$,
the number of irreducible representations of $\mathbb{Z}_r \wr S_n$ whose
determinant is $\zeta$.
