Rubik's Cube group

We consider two subgroups of G: First the subgroup C_o of cube orientations, the moves that leave the position of every block fixed, but can change the orientations of blocks. This group is a normal subgroup of G. It can be represented as the normal closure of some moves that flip a few edges or twist a few corners. For example, it is the normal closure of the following two moves:

${\displaystyle BR^{\prime }D^{2}RB^{\prime }U^{2}BR^{\prime }D^{2}RB^{\prime }U^{2},\,\!}$ (twist two corners)

${\displaystyle RUDB^{2}U^{2}B^{\prime }UBUB^{2}D^{\prime }R^{\prime }U^{\prime },\,\!}$ (flip two edges).

Second, we take the subgroup ${\displaystyle C_{P}}$ of cube permutations, the moves which can change the positions of the blocks, but leave the orientation fixed. For this subgroup there are several choices, depending on the precise way you define orientation.[note 2] One choice is the following group, given by generators (the last generator is a 3 cycle on the edges):

${\displaystyle C_{p}=[U^{2},D^{2},F,B,L^{2},R^{2},R^{2}U^{\prime }FB^{\prime }R^{2}F^{\prime }BU^{\prime }R^{2}].\,\!}$

Since C_o is a normal subgroup and the intersection of C_o and C_p is the identity and their product is the whole cube group, it follows that the cube group G is the semi-direct product of these two groups. That is

${\displaystyle G=C_{o}\rtimes C_{p}.\,}$

Next we can take a closer look at these two groups. The structure of C_o is

${\displaystyle \mathbb {Z} _{3}^{7}\times \mathbb {Z} _{2}^{11},\ }$

since the group of rotations of each corner (resp. edge) cube is ${\displaystyle \mathbb {Z} _{3}}$ (resp. ${\displaystyle \mathbb {Z} _{2}}$), and in each case all but one may be rotated freely, but these rotations determine the orientation of the last one. Noticing that there are 8 corners and 12 edges, and that all the rotation groups are abelian, gives the above structure.

Cube permutations, C_p, is a little more complicated. It has the following two disjoint normal subgroups: the group of even permutations on the corners A₈ and the group of even permutations on the edges A₁₂. Complementary to these two subgroups is a permutation that swaps two corners and swaps two edges. It turns out that these generate all possible permutations, which means

${\displaystyle C_{p}=(A_{8}\times A_{12})\,\rtimes \mathbb {Z} _{2}.}$

Putting all the pieces together we get that the cube group is isomorphic to

${\displaystyle (\mathbb {Z} _{3}^{7}\times \mathbb {Z} _{2}^{11})\rtimes \,((A_{8}\times A_{12})\rtimes \mathbb {Z} _{2}).}$

This group can also be described as the subdirect product

${\displaystyle [(\mathbb {Z} _{3}^{7}\rtimes \mathrm {S} _{8})\times (\mathbb {Z} _{2}^{11}\rtimes \mathrm {S} _{12})]^{\frac {1}{2}}}$,

in the notation of Griess.

When the centre facet symmetries are taken into account, the symmetry group is a subgroup of

${\displaystyle [\mathbb {Z} _{4}^{6}\times (\mathbb {Z} _{3}^{7}\rtimes \mathrm {S} _{8})\times (\mathbb {Z} _{2}^{11}\rtimes \mathrm {S} _{12})]^{\frac {1}{2}}.}$

(This unimportance of centre facet rotations is an implicit example of a quotient group at work, shielding the reader from the full automorphism group of the object in question.)

The symmetry group of the Rubik's Cube obtained by disassembling and reassembling it is slightly larger: namely it is the direct product

${\displaystyle \mathbb {Z} _{4}^{6}\times (\mathbb {Z} _{3}\wr \mathrm {S} _{8})\times (\mathbb {Z} _{2}\wr \mathrm {S} _{12}).}$

The first factor is accounted for solely by rotations of the centre pieces, the second solely by symmetries of the corners, and the third solely by symmetries of the edges. The latter two factors are examples of generalized symmetric groups, which are themselves examples of wreath products. (There is no factor for re-arrangements of the center faces, because on virtually all Rubik's Cube models, re-arranging these faces is impossible with a simple disassembly.)

The simple groups that occur as quotients in the composition series of the standard cube group (i.e. ignoring centre piece rotations) are ${\displaystyle A_{8}}$, ${\displaystyle A_{12}}$, ${\displaystyle \mathbb {Z} _{3}}$ (7 times), and ${\displaystyle \mathbb {Z} _{2}}$ (12 times).

It has been reported that the Rubik's Cube Group has 81,120 conjugacy classes.[8] The number was calculated by counting the number of even and odd conjugacy classes in the edge and corner groups separately and then multiplying them, ensuring that the total parity is always even. Special care must be taken to count so-called parity-sensitive conjugacy classes, whose elements always differ when conjugated with any even element versus any odd element.[9]