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
Prince Rupert's cube for the largest square that lies within a unit cube. More generally, show how to find the largest rectangle of a given aspect ratio that lies within a unit cube. As they show, the optimal rectangle must always pass through the center of the cube, with its vertices on edges of the cube. Based on this, they show, depending on the desired aspect ratio, that the optimal rectangle must either lie on a plane that cuts diagonally through four corners of the cube, or it must be formed by an isosceles right triangle on one corner of the cube and