For even-genus Riemann surfaces the largest purely non-free finite group action has order at least 8g and achieves equality infinitely often; for odd genus the order satisfies 4g ≤ μ(g) < 8g.
Bogopolski, Classifying of the actions of finite groups on orientable surfaces of genus 4, Siberian Advances in Mathematics,7(4) (1997) 9-38
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On the biggest purely non-free conformal actions on compact Riemann surfaces and their asymptotic properties
For even-genus Riemann surfaces the largest purely non-free finite group action has order at least 8g and achieves equality infinitely often; for odd genus the order satisfies 4g ≤ μ(g) < 8g.