I believe that I have come up with formulae for the groups representing the $2\times 2\times n$ Rubik's "cubes," but I need someone to confirm that they are correct. Here are the groups that I came up with for the groups in this form (they do not take pure rotations of the cube into account):
- For a $2\times 2\times 2$ cube, the associated group is $\mathbb Z_3^7\rtimes S_8$.
- For a $2\times 2\times (2n+1)$ cube, where $n\ge 1$, the associated group is $S_8^n\times S_4$.
- For a $2\times 2\times 2n$ cube, where $n\ge 2$, the associated group is $S_8^n$.
Can anyone confirm these results, if only for certain small values of $n$?