gt.geometric topology - Periodic mapping classes of the genus two orientable surface

In the paper listed below there is a calculation of all the finite group actions on a genus 2 surface. There are 20 of them, with the groups ranging from order 2 to order 48. Nine of the actions are of cyclic groups, of orders 2,2,3,4,5,6,6,8,10 respectively. The paper also does the genus 3 case. The techniques are mostly algebraic. It is an interesting exercise to try to find nice geometric pictures of all the actions.

S.A.Broughton, Classifying finite group actions on surfaces of low genus, J.Pure Appl.Alg. 69 (1991), 233-270.