= Rotational symmetry group of a regular icosahedron
The orientation-preserving <rotational symmetry> group of a <regular icosahedron> has order sixty. Its <conjugacy classes> have sizes $1,15,20,12,12$: identity; half-turns about opposite-edge axes; <rotations> through $\pm2\pi/3$ about opposite-face axes; and <rotations> through $\pm2\pi/5$ and $\pm4\pi/5$ about opposite-vertex axes, respectively. Conjugation carries an axis to its rotated axis and preserves the <rotation> angle. The <transitive group action> on edges, faces and vertices gives the indicated classes, with reversal of the axis identifying the two signs. A <normal subgroup> must contain whole <conjugacy classes>; none of the proper nonidentity sums of these class sizes is a divisor of sixty. Thus this group is a <simple group>.
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