Number of colorings of cube's faces
How many ways are there to color faces of a cube with N colors if two colorings are the same if it's possible to rotate the cube such that one coloring goes to another?
Solution 1:
The number of different colorings is equal to
\begin{equation*} \frac{n^6 + 3n^4 + 12n^3 + 8n^2}{24}. \end{equation*}
You can get this number using Burnside lemma. The wikipedia article contains solution of your problem as well.