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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?

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The number of different colorings is equal to $\frac{n^6 + 3n^4 + 12n^3 + 8n^2}{24}$.

You can get this number using Burnside lemma. The wikipedia article contains solution of your problem as well.

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