Template:Full octahedral group; single elements; signed perm mat, perm mat and cube

Description

[[File:Cube permutation {{{1}}} {{{2}}} JF.png|thumb|Corresponding permutation of the JF compound]] [[File:Cube permutation {{{1}}} {{{2}}} JF ortho.png|thumb|Orthographic projection shown in the file]] One of the 48 elements of the full octahedral group represented as

3×3 signed permutation matrix, the transformation matrix of the linear map (shown as text on the left)
8×8 permutation matrix (multiplication corresponds normal function composition)
permutation of the cube (the JF compound and the number cube on the left show the result)

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

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 $(m,n)=(m,0)\circ (0,n)$ means that first $(0,n)$ is applied and then $(m,0)$ .

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 $(m,0)$ with dark $m$ and $(0,n)$ with green $n$ 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.)