The full isometry group of a cube has order . Its transitive actions on the vertices, edges, and main diagonals have stabilizers isomorphic respectively toThe action on main diagonals has kernel .
The full symmetry group of a cube acts transitively and faithfully on its twelve edges. For vertices , its signed permutation matrices act transitively on the edges. The stabilizer subgroup of consists of independently swapping and reversing , and is a Klein four-group. The orbit-stabilizer theorem gives order . Fixing every edge fixes every vertex, as each vertex is the unique intersection of its three incident edges, so the group action is faithful. The resulting subgroup of has index and is not normal: central inversion has cycle type , whose conjugacy class in has elements.
The symmetry group of a cube splits into its central inversion and its rotational symmetry group of a cube. Inversion has determinant , the rotation subgroup has determinant , and together they generate the full group with trivial intersection. The rotation subgroup acts faithfully on the four body diagonals and is isomorphic to . Consequently the full group is this direct product of groups.
The orientation-preserving symmetries of a cube are the signed permutation matrices with determinant one. There are of them. Their group action on the four body diagonals identifies this group with the symmetric group . Indeed choose diagonal directions , 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 , which has determinant , so the kernel of a group homomorphism is trivial. Both groups have order 24, giving the identification. Its face permutations have cycle types , , , and , occurring times respectively.
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