Since "cubes" with higher dimension than three exist I think it's natural to ask for higher dimensional Rubik's cubes. These so called hypercubes don't seem to have been described from a group theoretic point of view.


Are there any papers on this? Is the group of the $3\times 3\times 3 \times 3$ cube a subgroup of a wreath product of another wreath product?


In case you don't know about the $3\times 3\times 3$ cube. Its group is a subgroup of a product of wreath products. The wreath products describing the corner pieces and the edge pieces and representing a permutation of them with its action in the respective orientation. That is why I conjecture that in the $3\times 3\times 3 \times 3$ case we might get a wreath product of the permutation of the faces, which are now 3d cubes itself, by the the wreath product of the permutation of the 2d faces of these by the groups of their orientations.



Higher dimensional Rubik's cube group
The 4-dimensional, i.e. $3 \times 3 \times 3 \times 3$, equivalent of the
Rubik's cube has 8 three-dimensional sides, each of which consists of
$3^3 = 27$ three-dimensional colored "squares".
Of these 27 "squares", the one in the center is fixed. Thus in total our
4-dimensional Rubik's cube has $8 \cdot 26 = 208$ movable "squares",
and the group of its sequences of moves under composition embeds therefore
into ${\rm S}\_{208}$.


Now just as the usual 3-dimensional Rubik's cube has 2 kinds of movable cubies
(edge stones and corner stones), our 4-dimensional Rubik's cube has 3 distinct
kinds of movable cubies:


* The *corner stones*, of which there are $2^4 = 16$.


Each corner stone has 4 visible colored "squares".
* The *edge stones*, of which there are $2 \cdot 12 + 8 = 32$.


Each edge stone has 3 visible colored "squares".
* The *face stones*, of which there are $2 \cdot 6 + 12 = 24$.


Each face stone has 2 visible colored "squares".


Therefore, analogous to the usual 3-dimensional case, we can finally say
that our 4-dimensional Rubik's cube group embeds into the following direct
product of wreath products:
$$
 {\rm A}\_4 \wr {\rm S}\_{16} \times {\rm S}\_3 \wr {\rm S}\_{32} \times
 {\rm C}\_2 \wr {\rm S}\_{24}.
$$


