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 :) 



Periodic mapping classes of the genus two orientable surface
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. 


