Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be
$Z/\ell Z$ semi-direct product $Z/pZ$ where $\ell$ is a prime distinct from
$p$. In this paper, we study Galois covers $\psi:Z \to P^1_k$ ramified only
over $\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$
that admits such a cover and show that it depends only on $\ell$, $p$, and the
order $a$ of $\ell$ modulo $p$. We also prove that the number of curves $Z$ of
this minimal genus which admit such a cover is at most $(p-1)/a$.