= Direct-product decomposition of the cube symmetry group
{title2=$G_{\mathrm{cube}}\cong C_2\times S_4$}
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>.
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