A group acting on a compact Riemann Surface (awaiting review)

A group $G$ acts on a compact Riemann surface $X$ if there is an injective homomorphism from $G$ to the group of automorphisms of $X$, i.e. a homomorphism $\phi: G \hookrightarrow \mathrm{Aut}(X)$.

Equivalently, a group $G$ acts on $X$ if there is a Fuchsian group $\Gamma$ and a surjective homomorphism $\eta: \Gamma \to G$ with $\ker( \eta)$ isomorphic to the fundamental group of $X$.