2007 AMC 12B Problem 16

Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?

\mathrm {(A)} 15\qquad \mathrm {(B)} 18\qquad \mathrm {(C)} 27\qquad \mathrm {(D)} 54\qquad \mathrm {(E)} 81

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.