Can anyone review my work on this problem and tell me if I'm missing anything major? Thanks!
Q: Describe the orbits of poles for the group of rotations of an octahedron.
There are $|G|=N=24$ rotational symmetries for an octahedron. These can be split up into three pole orbits for edges, faces, and vertices respectively. Using the book notation of $r_i$ for the size of the stabilization group and $n_i$ for the size of the orbit we have the following pole orbits
- Edges: $r_i = 2, n_i = 12$.
- Vertices: $r_i = 3, n_i = 8$.
- Faces: $r_i = 4, n_i = 6$.
There are indeed $24$ elements of the group. You should always ask yourself which one is the identity, however.