Semi-direct Galois covers of the affine line

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