Refined passport (reviewed)

Let $G$ be a group of automorphisms acting on a compact Riemann surface $X$ of genus $g$, let $(s_1, \ldots, s_n)$ be a generating vector for the action of $G$ on $X$, and let $\mathcal{C}=(C_1, \ldots, C_n)$ be the corresponding vector of conjugacy classes of $G$.

The tuple $(g,G,\mathcal{C})$ is called a refined passport (some authors say that $X$ is of ramification type $(g,G,\mathcal{C})$).