Recall that a regular icosahedron has 20 faces, 30 edges and 12 vertices. Let $G$ be the group of rotational symmetries of a regular icosahedron.

Compute the conjugacy classes of $G$. Hence, or otherwise, construct the character table of $G$. Using the character table explain why $G$ must be a simple group.

[You may use any general theorems provided that you state them clearly.]