The class equation of the octahedral group 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?
 A: 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.
A: 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]
  

A: Following the Wikipedia list of conjugacy classes given by Zev Chonoles, we obtain:
24 = 1 (identity from the whole octahedron) + 8 (rotation by 120° about a face, an axis with 3-fold symmetry) + 6 (rotation by 180° about an edge, an axis with 2-fold symmetry) + 6 (rotation by 90° about a vertex, an axis with 4-fold symmetry) + 3 (rotation by 180° about a vertex, an axis with 4-fold symmetry)
This is the exact relation between conjugacy classes and rotations about icosahedron's axes of symmetry
For details see Algebra 2nd edition chapter 7.4 'The Class Equation of the Icosahedral Group' by M. Artin
