File:Cube permutation 6 3.svg

One of the 48 elements of the full octahedral group represented as

The cube has the coordinate origin in its center. Vertex 0 has coordinate ${\displaystyle (-1,-1,-1)}$, vertex 1 has ${\displaystyle (1,-1,-1)}$, and vertex 7 has ${\displaystyle (1,1,1)}$.

Transformation matrices (3×3) in the same row have the same pattern of negative rows, and those in the same column have the same pattern of non-zero entries.

This is left action: Permutation ${\displaystyle (m,n)=(m,0)\circ (0,n)}$ means that first ${\displaystyle (0,n)}$ is applied and then ${\displaystyle (m,0)}$.

The 24 permutations of the cube that leave a contained tetrahedron unchanged (those where the bottom left vertex of the resulting cube is light gray) are identified with numbers 0..23 corresponding to elements of S₄. These elements' combinations with the inversion are identified with the same number with an added apostrophe.

Transformations ${\displaystyle (m,0)}$ with dark ${\displaystyle m}$ and ${\displaystyle (0,n)}$ with green ${\displaystyle n}$ are odd (have a negative determinant). Permutations in these rows and columns are marked with a horizontal and vertical gray bar respectively. (Conveniently, where it looks like a minus the determinant is negative, and when it looks like a plus it is positive.)