On Galois group of factorized covers of curves

Let $\mathcal{Y}\xrightarrow {{\psi}} \mathcal{X} \xrightarrow {\varphi} \mathbb{P}^{1}$ be a sequence of covers of compact Riemann surfaces. In this work we study and completely characterize the Galois group $\mathfrak{G}(\varphi\circ\psi)$ of $\varphi\circ\psi$ under the following properties: $\varphi$ is a simple cover of degree $m$ and $\psi$ is a Galois unramified cover of degree $n$ with abelian Galois group of type $(n_1,n_2,\dots,n_s)$. We prove that $ \mathfrak{G}(\varphi\circ\psi) \cong ({\mathbb Z}_{n_1} \times {\mathbb Z}_{n_2} \times \cdots \times {\mathbb Z}_{n_s})^{m-1} \rtimes {\bf S}_m$. Furthermore, we give a natural geometric generator system of $\mathfrak{G}(\varphi\circ\psi)$ obtained by studying the action on the compact Riemann surface $\mathcal{Z}$ associated to the Galois closure of $\varphi\circ\psi.$