By the Hurwitz's automorphisms theorem there is an upper bound $\text{Aut}(C)\leq 84(g1)$ for all Riemann surfaces $C$ with $g(C)\geq 2$, but it is not sharp if $g=2$. What is the sharp upper bound for genus $2$ Riemann surfaces? Also, I'm asking for the simple proof of the fact that there is no genus $2$ Riemann surfaces with $Aut(C)=84$.
One proof uses the fact that every genus $2$ Riemann surface is hyperelliptic. If $C$ is a hyperelliptic Riemann surface, then there is a unique degree $2$ map to $\mathbb P^1$, which defines a homomorphism $Aut(C) \to Aut(\mathbb P^1)$ whose kernel is of order $2$, generated by the hyperelliptic involution. This gives a finite subgroup of $Aut(\mathbb P^1) = PGL_2(\mathbb C)$ which preserves some set of exactly $2g+2$ points. A finite subgroup of $PGL_2(\mathbb C)$ must lie in its maximal compact subgroup, which is $SO(3)$. We can use the classification of finite subgroups of $SO(3)$ to find the possible automorphism groups. We need to find the largest finite subgroup of $SO(3)$ that preserves a set of $6$ points on the sphere. We use the fact that all subgroups are either cyclic, dihedral, $A_4$, $S_4$, or $A_5$. A cyclic subgroup that fixes $6$ points is no larger than $C_6$. A dihedral subgroup that fixes $6$ points is no larger than $D_6$. $A_5$ has no orbits of fewer than $12$ points. $S_4$ is the largest remaining possibility, and preserves $6$ points (the $6$ vertices of an inscribed regular octahedron), so the largest possible value for $Aut(C)$ is $2S_4=48$. 


It follows immediately from Hurwitz' theorem on the genus that a curve of genus 2 cannot have an automorphism of order 7, by considering the quotient map by this action. Hurwitz himself proved that if 84(g1) does not occur, the next largest possible bound is 48(g1). I.e. his method gives a list of possible bounds, not just one maximal bound. Mathematische Annalen 1892, Volume 41, Issue 3, pp 403442 This is offered as an argument for reading the original work. see also: 


http://amathew.wordpress.com/2011/11/10/automorphismsofcompactriemannsurfaces/#more2954 should be of interest to you. 

