Corresponding permutation of the JF compoundOrthographic projection shown in the fileOne of the 48 elements of the full octahedral group represented as


[[ 0, 1, 0], [ 1, 0, 0], [ 0, 0,-1]]


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){\displaystyle (-1,-1,-1)}, vertex 1 has (1,−1,−1){\displaystyle (1,-1,-1)}, and vertex 7 has (1,1,1){\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 (m,n)=(m,0)∘(0,n){\displaystyle (m,n)=(m,0)\circ (0,n)} means that first (0,n){\displaystyle (0,n)} is applied and then (m,0){\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 (m,0){\displaystyle (m,0)} with dark m{\displaystyle m} and (0,n){\displaystyle (0,n)} with green n{\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.)

The 48 permutations in a 8x6 matrix
The right-handed coordinate system used
Original position of the JF compoundOriginal position of the cube
Own work
WatchduckYou can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck".