The Rubik's Cube group ( G , ⋅ ) {\displaystyle (G,\cdot )} represents the mathematical structure of the Rubik's Cube mechanical puzzle. Each element of the set G {\displaystyle G} corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces. With this representation, not only can any cube move be represented, but any position of the cube as well, by detailing the cube moves required to rotate the solved cube into that position. 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 G {\displaystyle G} . The group operation ⋅ {\displaystyle \cdot } is the composition of cube moves, corresponding to the result of performing one cube move after another. 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. Using this representation, the solved cube is the identity permutation which leaves the cube unchanged, while the twelve cube moves that rotate a layer of the cube 90 degrees are represented by their respective permutations. The Rubik's Cube group is the subgroup of the symmetric group S 48 {\displaystyle \mathrm {S} _{48}} generated by the six permutations corresponding to the six clockwise cube moves. With this construction, any configuration of the cube reachable through a sequence of cube moves is within the group. Its operation ⋅ {\displaystyle \cdot } refers to the composition of two permutations; within the cube, this refers to combining two sequences of cube moves together, doing one after the other. The Rubik's Cube group is non-abelian as composition of cube moves is not commutative; doing a sequence of cube moves in a different order can result in a different configuration.
Cube moves A 3 × 3 × 3 {\displaystyle 3\times 3\times 3} Rubik's Cube consists of 6 {\displaystyle 6} faces, each with 9 {\displaystyle 9} colored squares called facelets, for a total of 54 {\displaystyle 54} facelets. A solved cube has all of the facelets on each face having the same color. A cube move rotates one of the 6 {\displaystyle 6} faces either 90 ∘ , 180 ∘ , {\displaystyle 90^{\circ },180^{\circ },} or − 90 ∘ {\displaystyle -90^{\circ }} (half-turn metric). A center facelet rotates about its axis but otherwise stays in the same position. Cube moves are described with the Singmaster notation:
The empty move is E {\displaystyle E} . The concatenation L L L L {\displaystyle LLLL} is the same as E {\displaystyle E} , and R R R {\displaystyle RRR} is the same as R ′ {\displaystyle R^{\prime }} .
Group structure
The following uses the notation described in How to solve the Rubik's Cube. The orientation of the six centre facelets is fixed. 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, { F , B , U , D , L , R } {\displaystyle \{F,B,U,D,L,R\}} . The cardinality of G is given by:
| G | = 43,252,003,274,489,856,000 {\displaystyle |G|=43{,}252{,}003{,}274{,}489{,}856{,}000}
= 12 ! ⋅ 8 ! 2 ⋅ 2 12 2 ⋅ 3 8 3 {\displaystyle {}={\frac {12!\cdot 8!}{2}}\cdot {\frac {2^{12}}{2}}\cdot {\frac {3^{8}}{3}}}
= 2 27 3 14 5 3 7 2 11 {\displaystyle {}=2^{27}3^{14}5^{3}7^{2}11}
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 (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). The largest order of an element in G is 1260. For example, one such element of order 1260 is
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