All symmetry groups for surfaces of genus 2 and 3 are known. In this paper, we classify symmetry groups and tiling groups with three branch points for surfaces of genus 4 and 5. Also, a class of symmetry groups that are not tiling groups is presented, as well as a class of odd order non-abelian tiling groups.
Vinroot, C. Ryan
"Symmetry and Tiling Groups for Genus 4 and 5,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 1
, Article 5.
Available at: http://scholar.rose-hulman.edu/rhumj/vol1/iss1/5