Rubik's Cube group

The following uses the notation described in How to solve the Rubik's Cube. The orientation of the six centre facets is fixed.

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[7]).

The largest order of an element in G is 1260. For example, one such element of order 1260 is

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:

Since Co is a normal subgroup and the intersection of Co and Cp 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

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

Cube permutations, Cp, 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

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

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. The latter two factors are examples of generalized symmetric groups, which are themselves examples of wreath products.

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]