Periodic mapping classes of the genus two orientable surface Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and reintrepreting them as a circle bundles over orbifolds. 
In the http://web.archive.org/web/20070316045651/http://www.smm.org.mx/SMMP/html/modules/Publicaciones/AM/Cm/35/artExp08.pdf -work you would like see the cases $O_1$, among $N_1$ and $N_2$, solved. Any feedback on the results and conjectures, some of them obviously false, will bring a lot of happiness :) 
 A: If you want to enumerate the finite-order automorphisms (up to conjugacy) I suggest the following exercise.  The associated 3-manifold is Seifert fibred.  So determine how the genus 2 surface is sitting in the Seifert manifold (horizontal incompressible surface).  
This will give you a formula relating the various branch points of the monodromy to the Seifert data.  Moreover, you should be able to go back-and-forth between the description of the Seifert-fibred space (unnormalized Seifert data, fibred over a genus 0, 1 or 2 surface) and the monodromy of the surface.   So the classification of Seifert-fibred spaces basically gives you a dictionary for walking-through the finite-order automorphisms of a mapping class group. 
A: 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.
