Rotational symmetry group of a cube ($|H|=24$)

The orientation-preserving symmetries of a cube are the $3\times 3$ signed permutation matrices with determinant one. There are $3!2^2=24$ of them. Their group action on the four body diagonals identifies this group with the symmetric group $S_4$. Indeed choose diagonal directions $(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)$, whose sum is zero and whose only linear relation has all coefficients equal. A rotation fixing every diagonal sends each direction to itself or its negative; the relation forces all four signs equal. All negative would give $-I$, which has determinant $-1$, so the kernel of a group homomorphism is trivial. Both groups have order 24, giving the identification. Its face permutations have cycle types $1^6$, $1^{2}4$, $1^2{2}^2$, $2^3$ and $3^2$, occurring $1,6,3,6,8$ times respectively.

For cube vertices $(\pm 1,\pm 1,\pm 1)$, the four vertices of coordinate product $+1$ and the four of product $-1$ form the two inscribed regular tetrahedra. A cube rotation is a signed permutation matrix of determinant one and maps each product class to one class. A quarter-turn about a coordinate axis exchanges the classes. The orbit of either tetrahedron therefore has size two, while the cube rotation group has order 24. The orbit-stabilizer theorem gives its setwise stabilizer subgroup order 12; this is the tetrahedron's proper rotation group.

Permuting the three pairs of opposite faces gives a group homomorphism from the rotational symmetry group of a cube to $S_3$. Its kernel consists of the identity and the half-turns about the three coordinate axes: these are precisely the diagonal sign matrices with determinant one. Thus it is a normal subgroup of order four, a Klein four-group.

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.