/sci/ - Science & Math

Need some help with combinatorics. I was looking at an old toy which is a dodecahedron with rounded magnetic blocks for faces. There are three blocks of each four different colours red, green, blue and yellow. I wanted to calculate the number of patterns you could create by rearranging the blocks, assuming rotations and colours can be interchanged. The way I went about it is this:
[math]12![/math] ways to arrange the faces
[math]12\cdot 5[/math] ways to rotate since every rotation is defined by a face and one of its 5 neighbours
[math]4![/math] ways to rearrange the 4 colors
[math]3!^4[/math] ways to rearrange the 3 faces for each of the 4 colour groups
[eqn]\frac{12!}{(12\cdot 5)3!^4 4!}=\frac{770}{3}[/eqn]
Since the result is not an integer, I suspect I fucked up somewhere. Any ideas?