Finite group actions on compact Riemann surfaces are a classical subject, and the related literature is huge.
It is well known that if a finite group $G$ acts as a group of automorphisms on a compact Riemann surface of genus $g$, g \geq 2$, then necessarily
$|G| \leq 84(g-1)$.
This is a old result of Hurwitz, and if equality holds then the group $G$ is called a Hurwitz group in genus $g$. The classification of Hurwitz groups is not yet completed; it is known that there exists a Hurwitz group for infinitely many values of $g$, and that there exists no Hurwitz group for infinitely many values of $g$ as well.
Moreover, any Hurwitz group $G$ is a quotient of the infinite triangle group
$T_{2,3,7}=\langle x, y | x^2=y^3=(xy)^7=1 \rangle$.
There exist no Hurwitz group in genus $2$, and exactly one in genus $3$. It is the group $G=PSL(2, \mathbb{F}_7)$, the unique simple group of order $168$. The corrisponding Riemann surface can be realized as a particular curve of degree $4$ in $\mathbb{P}^3(\mathbb{C})$, the so-called Klein quartic.
