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. 1 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$g$, 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.