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Has anyone made a series of Rubik's cubes showing all the different subgroups of the big permutation group (for some value of "all")?
Like, the trivial cube would have the same plain colour on every face, and the biggest group cube would have unique, oriented stickers on each face.
And in between, there'd be a cube with all the corners the same, but different middle faces, for example.
now that you can get really cheap Rubik's cubes as promo items, somebody needs to stop me actually doing this
@christianp If I understand it correctly, the Rubik's Cube group has 81120 conjugacy classes (per Wikipedia), therefore you might need to make a lot of cubes. But maybe I'm doing the group theory wrong?
Also they're mostly products of edge and corner classes separately, so perhaps it's more manageable to have separate edge- and corner-decorated cubes.
@Colinvparker yes, that's why I said for some value of "all" 
I think picking a few examples would be enough to get the gist, maybe picking a few chains of subgroups 𝑆ₙ such that 1<𝑆₁<…<𝑆ₖ<𝐺, where 𝐺 is the whole Rubik's cube group.
@christianp Yeah, I would still pick examples where either the edges or corners were trivial, which simplifies things a lot and likely provides a better illustration.
@Colinvparker @christianp conveniently, the subgroup of cubie-orientation-preserving permutations decomposes into a semidirect product of (permutations of corners), (permutations of edges) and (a permutation of exactly two corners and two edges).