I want to solve the following:
Let $I$ be the group of isometries that preserve the orientation of the icosahedron. Use the class formula to show that $I$ is a simple group (i.e. It does not have proper normal subgroups) of order 60.
Well, the thing is that I dont know how to use the class formula, I have checked How to prove that a group is simple only from its class equation, but I dont know if it works in my case. Can you help me to prove this please?
Thanks a lot in advance.
My class formula:
$|G|= |Z(G)|+ \sum_{g \in R´} [G:Z(g)]$
where $Z(g):= G_g$ the isotropy group given by the action conjugation. And $R´$ is the set of representatives of all te non-trivial conjugacy classes