TITLE: Automorphisms of genus 6 surfaces
QUESTION [14 upvotes]: I have found in Wikipedia that the number of automorphisms of a Riemann surface of genus 6 does not exceed 150. The page offers neither a proof nor a reference. Can someone help me?

REPLY [3 votes]: The question is purely combinatorial - the Hurwitz bound comes from the observation that the quotient of a surface by its automorphism group is a hyperbolic orbifold, and the hyperbolic orbifold of smallest area is the $(2, 3, 7)$ triangle orbifold, Dividing the area of your surface by the area of the orbifold gives you the Hurwitz bound. However, for some genera, there is no triangulation where the vertices have the right degrees, then you have to look for bigger triangle groups. For the surface of genus $6,$ you get the $(2, 3, 10)$ triangle group, but in any case, this is just a combinatorial exercise (for any fixed genus).