G-set counting(2),Tetrahedron

aliceinwonderland · #1 $Old$ December 15th, 2008, 05:55 PM

Wooden tetrahedrons of the same size are to be painted on each face. How many tetrahedrons can be made if eight colors of paint are available? Colors may be repeated on different faces at will. Define G for rigid motions and use Burnside's formula.

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My attempt to this problem so far

|G| = 12

One identity element which leaves all $8^{4}$ elements of X unchanged.
Eight rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together 8, each of which leaves $8^{2}$ unchanged.

If we give numbers like (bottom 1, left 2, right 3, back 4), a rotation by an angle of 180° such that an edge maps to the opposite edge (link) gives a permutation of sides (12)(34) for the linked picture.
Three actions for above can be defined, each of which leaves $8^{2}$ unchanged.

Thus, the number of distinguishable tetrahedrons are
$\frac{8^{4} + 8 \cdot 8^{2} + 3 \cdot 8^{2}}{12}$

Is this correct?