The class equation of the octahedral group

Question

I know that the class equation of the octahedral group is this: $$1 + 8 + 6 + 6 + 3$$

I think the $8$ stands for the $8$ vertices, the $6$ could be $6$ faces and $6$ pairs of edges. Then what is the $3$ for?

Answer 1

These numbers have nothing to do with the numbers of vertices, faces, or edges, at least not so directly. The octohedral group has $1$ identity element, which is the sole member of its own conjugacy class. The group has $8$ three-fold rotations along axes joining two opposite vertices. It has $6$ two-fold rotations along axes joining the midpoints of two opposite edges. It has $6$ four-fold rotations along axes joining the midpoints of opposite faces. Finally, it has $3$ two-fold rotations along axes joining the midpoints of opposite faces.

Answer 2

As described on the Wikipedia page for the octahedral group, the conjugacy classes are

• identity
• 6 × rotation by 90° about [an axis with 4-fold symmetry]
• 8 × rotation by 120° about a [an axis with 3-fold symmetry]
• 3 × rotation by 180° about a [an axis with 4-fold symmetry]
• 6 × rotation by 180° about a [an axis with 2-fold symmetry]