= Edge colourings of a tetrahedron under rotations
{title2=$\frac{q^6+8q^2+3q^4}{12}$}
The twelve rotational symmetries of a <tetrahedron> act on its six edges. Their edge cycle patterns are six fixed edges for the identity, two three-cycles for each of eight <rotational symmetries> of order three, and two fixed edges plus two transpositions for each of three half-turns. <Burnside's lemma> therefore counts <edge colourings> with $q$ labelled colours by $(q^6+8q^2+3q^4)/12$.
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