Face-pair orbits of cube rotations
= Face-pair orbits of cube rotations
{title2=$|\mathcal O|=6,6,24$}
The <rotational symmetry group of a cube> has three <group orbits> on ordered pairs of faces: equal faces, opposite faces, and adjacent faces. Their sizes are respectively $6$, $6$ and $24$. Fixing one face leaves its cyclic group of four quarter-turns, which acts transitively on the four adjacent faces; this proves transitivity in the last orbit rather than merely counting its points.