Coloring a Cube

Coloring each face a solid color and using each of N colors (N=2,3,4,5,6)
at least once,
in how many ways can you color a cube, such that a duplicate cannot be
found simply by rotating the cube?

Using each of N colors at least once (N=2,3,4), in how many ways
can you color a tetrahedron?
The "Using each of N colors at least once" in the problem statements
helps to limit the number of solutions. Can you solve them if that
phrase is changed to "Using N colors"?
Source: Several sources.
Solution
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