[kjwijicq] C6 puzzle

part of the key idea at the core of the Kociemba algorithm for Rubik's cube is that the cyclic group over 4 items has as a subgroup the cyclic group over 2 items.

this suggests a puzzle which has interleaved cyclic groups over 6 items, with subgroups of 2 and 3 somehow making simpler puzzles.  devil in the details.

unfortunately, no polyhedron with all regular hexagon faces exists, so we need to be more clever.  perhaps some regular hexagon faces: several with this property exist among Archimedean solids, e.g., truncated tetrahedron, a Pyraminx without its tips.  the Skewb cross section of a cube a regular hexagon, though, in a Skewb, only 3 of the 6 rotations of the hexagon are used.  the regular hexagon tiling in a toroidal universe.

general idea is a puzzle which rewards understanding of group theory.  cyclic group of 6 might be too obvious to learn generalities.