TITLE: Automorphisms of Riemann Surfaces
QUESTION [15 upvotes]: Izumi Kuribayashi and Akikazu Kuribayashi have classified all groups of automorphisms of compact Riemann surfaces of genus 3,4,5. (J. Pure Applied Alg.65(3)-Sept.1990, and J. Alg.134(1) Oct.1990) 
What are the other values of $g\geq 6$, for which a complete classification of automorphisms of compact Riemann surfaces of genus $g$ has been obtained? Can one suggest references for it, please?

REPLY [7 votes]: While this question was posed several years ago, I wanted to add one reference which was not mentioned above.
Marston Conder has computed lists of all “large” automorphism groups acting on curves up to genus 101, where by “large” we mean that if $G$ acts on a curve $X$ of genus $g$, then $|G|>4(g-1).$ (This means the genus of the quotient $X/G$ must be $0$ and there are at most $4$ branch points in the mapping $X \to X/G$.) His data is here:
https://www.math.auckland.ac.nz/~conder/BigSurfaceActions-Genus2to101-ByGenus.txt
He presents his groups as a list of generators so that the group is the quotient of the free group on  $3$ or $4$ variables (depending on the number of branch points) by those generators. 
Additionally, Thomas Breuer's data can be a bit hard to find.  Since I regularly need the data for my own research, I put his full list of groups and signatures up to genus 48 here:
http://www.math.grinnell.edu/~paulhusj/Monodromy/groupsignaturedata
Each line of that file reads as follows:
[*genus, order of group, signature, group identification number *]

where the group identification number is a tuple $(a,b)$ where the group $G$ is SmallGroup(a,b) in the database of small groups in GAP or Magma.