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now that you can get really cheap Rubik's cubes as promo items, somebody needs to stop me actually doing this

@christianp I think I'd actually flip your idea, Galois-connection stylee, and have each cube be invariant under the subgroup it represents. So the {e, superflip} cube has the usual face and corner pieces, and each edge is a rotation-invariant pattern of the two colours that normally live on that edge.

@pozorvlak is a cube representing $G/S$ in your scheme not the same as a cube representing 𝑆 in my scheme?
(I have forgotten lots of group theory)

@christianp I have also forgotten lots of group theory, but you can only take quotients by normal subgroups so I don't think the schemes are equivalent. Hrm. I need to think about this more.

@pozorvlak that's the wrinkle I was reaching for!

@pozorvlak I wonder if this is feasible to do online, before making a physical version

@christianp I'd definitely want to at least enumerate the subgroups before I started buying cubes! Do you know how many there are? Wikipedia gives the structure as $({\mathbb{Z}}_3^7\times {\mathbb{Z}}_2^{11})⋊((A_8\times A_{12})⋊{\mathbb{Z}}_2)$ - if you went down that far you'd want ten cubes (the whole group, ${\mathbb{Z}}_3^7\times {\mathbb{Z}}_2^{11}$, ${\mathbb{Z}}_3^7$, etc), but most of those components have lots of subgroups of their own.

@pozorvlak oh, there are thousands and thousands, so you could only hope to pick a sample of them

@christianp I dunno, "I filled a school hall full of suckers^W volunteers painting tiny Rubik's cubes to elucidate the subgroup structure" sounds very much in @standupmaths's wheelhouse...

@jgamble feels like a project for @standupmaths

@christianp @jgamble I'm doing product development for Maths Gear now so it'll happen if @standupmaths doesn't specifically stop me from doing it

@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).