Direct-product decomposition of the cube symmetry group ($G_{\mathrm{cube}}\cong C_2\times S_4$)

The symmetry group of a cube splits into its central inversion and its rotational symmetry group of a cube. Inversion has determinant $-1$, the rotation subgroup has determinant $1$, and together they generate the full group with trivial intersection. The rotation subgroup acts faithfully on the four body diagonals and is isomorphic to $S_4$. Consequently the full group is this direct product of groups.