The Monodromy Group of pq-Covers

In this work we study the monodromy group of covers $\varphi \circ \psi$ curves \linebreak $\mathcal{Y}\xrightarrow {\quad {\psi}} \mathcal{X} \xrightarrow \varphi} \mathbb{P}^{1}$, where $\psi$ is a $q$-fold cyclic \'etale cover and $\varphi$ totally ramified $p$-fold cover, with $p$ $q$ different prime numbers odd. We show that Galois $\mathcal{G}$ closure $\mathcal{Z}$ form $ \mathcal{G} = \mathbb{Z}_q^s \rtimes \mathcal{U}$, $0 \leq s p-1$ $\mathcal{U}$ simple transitive permutation degree $p$. Since are known, construct examples such these groups, result very from previously known case when was assumed to be cyclic, in which \mathbb{Z}_p$. Furthermore, able characterize subgroups $\mathcal{H}$ $\mathcal{N}$ $\mathcal{Y} \mathcal{Z}/\mathcal{N}$ $X \mathcal{Z}/\mathcal{H}$.