1. How many different tetrahedrons can be produced by coloring each face a solid color and using n different colors? (Two tetrahedrons are the same if they can be turned and placed side by side so that corresponding sides match in color.)

2. How many cubes with n colors?

I sneaked a peek at Charlie's tetrahedron answer (just the answer), and though it looks completely different, it simplifies to my answer.  That's good, but the cube will be much harder, and much more likely to have an error.
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I'm encountering difficulties with the cube.  I don't have any thorough method of counting all the rotational groups.  Using the same method for solving part 2 will be much harder, and I'm searching for a more elegant method.