Number of ways to color cube sides.

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How many ways can cube sides be colored by 4 types of color? I need something, not exactly the answer, maybe just formulas or theory I need. Up to 4 different colors. no, rotation does not change.

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This is a case for Burnside's Lemma, AKA Polya counting theory. The Wikipedia article explains exactly your problem in detail, albeit with three colors instead of four. Now translate the described procedure to your situation.