Rubik's Cube group 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 wreath products. The simple groups that occur as quotients in the composition series of the standard cube group (i.e. ignoring centre piece rotations) are formula_37, formula_38, formula_30 (7 times), and formula_31 (12 times). Rubik's Cube group The Rubik’s Cube group is a group formula_1 that represents the structure of the Rubik's Cube mechanical puzzle. Each element of the set formula_2 corresponds to a cube move, which is the effect of
Cube faces and eight triangular ones. In particular we can get regular octagons (truncated cube). The rhombicuboctahedron is obtained by cutting off both corners and edges to the correct amount. A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes. If two opposite corners of a cube are truncated at the depth of the three vertices directly connected to them, an irregular octahedron is obtained.