Theorem:
Some large natural symmetry groups of the sets of 8, 16, 32, or 64 points
in Euclidean space that are located at the vertices of a cube in 3, 4, 5. or 6
dimensions are generated by, respectively, arbitrary permutations of
parallel edges or parallel faces or parallel cubes or parallel hypercubes .
(For an example, see Diamond Theory in 1937.)
Illustration of related group actions:
