Face-pair orbits of cube rotations ($|\mathcal{O}|=6,6,24$)

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.