Rubik's Cube group

represents the structure of the Rubik's Cube mechanical puzzle.

Indeed with the solved position as a starting point, there is a one-to-one correspondence between each of the legal positions of the Rubik's Cube and the elements of

The Rubik's Cube is constructed by labeling each of the 48 non-center facets with the integers 1 to 48.

Each configuration of the cube can be represented as a permutation of the labels 1 to 48, depending on the position of each facet.

generated by the six permutations corresponding to the six clockwise cube moves.

colored squares called facelets, for a total of

A solved cube has all of the facelets on each face having the same color.

[3] A center facelet rotates about its axis but otherwise stays in the same position.

We can identify each of the six face rotations as elements in the symmetric group on the set of non-center facelets.

More concretely, we can label the non-center facelets by the numbers 1 through 48, and then identify the six face rotations as elements of the symmetric group S48 according to how each move permutes the various facelets.

The Rubik's Cube group, G, is then defined to be the subgroup of S48 generated by the 6 face rotations,

The cardinality of G is given by:[5] Despite being this large, God's Number for Rubik's Cube is 20; that is, any position can be solved in 20 or fewer moves[3] (where a half-twist is counted as a single move; if a half-twist is counted as two quarter-twists, then God's number is 26[6]).

[2] The center of G consists of only two elements: the identity (i.e. the solved state), and the superflip.

We consider two subgroups of G: First the subgroup Co 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: Second, we take the subgroup

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 'orientation' is defined.

The structure of Co is since the group of rotations of each corner (resp.

Noticing that there are 8 corners and 12 edges, and that all the rotation groups are abelian, gives the above structure.

It turns out that these generate all possible permutations, which means Putting all the pieces together we get that the cube group is isomorphic to This group can also be described as the subdirect product in the notation of Griess[citation needed].

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

(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[citation needed].)

The simple groups that occur as quotients in the composition series of the standard cube group (i.e. ignoring centre piece rotations) are

It has been reported that the Rubik's Cube Group has 81,120 conjugacy classes.

[7] 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.