13-piece puzzle

The puzzle has a dodecahedron (12-sided regular polyhedron) in the middle, with magnets on all sides. The other 12 pieces are shaped like a cone with the convex round base, and have each a piece of metal at the apex so they click nicely to the dodecahedron. When the puzzle is assembled, the whole thing looks a lot like a sphere. There are probably pictures of it somewhere, I got mine from Ikea ages ago.

Of the 12 cones, three have a red base, three have blue, three yellow, and three green base.

How many ways are there to arrange the 12 pieces around the magnet are there, when arrangements are considered different if the can't be made an exact match without detaching a piece? Cones with the same color base are considered identical.

(that is, a two six-sided dice, one with a 1 face on top, then clockwise 2-3-4-5 under it and 6 on the bottom face is the same as 6 on top, 3-2-5-4 clockwise under it, and 1 on the bottom, as one can be made the other by flipping it upside down and turning it a bit; but it is different from 1 on top, 2 on bottom, 3-5-4-6 clockwise in between).

Solution.
The number of ways of arranging 3 red, 3 blue, 3 yellow, 3 green objects is 12!/(3!x3!x3!x3!) = 369600 where, for an example, 7! = 1x2x3x4x5x6x7. Then you have to divide this by the number of ways a dodecahedron can be turned into different positions. This is 12x5 = 60. So the answer is 369600/60 = 6160.

Yep =)