The group of Rubik's Cube (TM) is a group of cube transformations of the popular Rubik's Cube (TM) puzzle game.

The RubiksCubeGroup() command returns a permutation group isomorphic to the Rubik's Cube (TM) group.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$R≔\mathrm{RubiksCubeGroup}\left(\right)$

$R≔⟨\left(6\,25\,43\,16\right)\left(7\,28\,42\,13\right)\left(8\,30\,41\,11\right)\left(17\,19\,24\,22\right)\left(18\,21\,23\,20\right)\,\left(1\,14\,48\,27\right)\left(2\,12\,47\,29\right)\left(3\,9\,46\,32\right)\left(33\,35\,40\,38\right)\left(34\,37\,39\,36\right)\,\left(1\,17\,41\,40\right)\left(4\,20\,44\,37\right)\left(6\,22\,46\,35\right)\left(9\,11\,16\,14\right)\left(10\,13\,15\,12\right)\,\left(3\,38\,43\,19\right)\left(5\,36\,45\,21\right)\left(8\,33\,48\,24\right)\left(25\,27\,32\,30\right)\left(26\,29\,31\,28\right)\,\left(1\,3\,8\,6\right)\left(2\,5\,7\,4\right)\left(9\,33\,25\,17\right)\left(10\,34\,26\,18\right)\left(11\,35\,27\,19\right)\,\left(14\,22\,30\,38\right)\left(15\,23\,31\,39\right)\left(16\,24\,32\,40\right)\left(41\,43\,48\,46\right)\left(42\,45\,47\,44\right)⟩$

$\mathrm{GroupOrder}\left(R\right)$

$43252003274489856000$

$\mathrm{IsSimple}\left(R\right)$

$\mathrm{false}$

$\mathrm{Degree}\left(R\right)$

$48$

$\mathrm{IsTransitive}\left(R\right)$

$\mathrm{false}$

$\mathrm{numelems}\left(\mathrm{Orbits}\left(R\right)\right)$

$2$

$\mathrm{map}\left(\mathrm{numelems}\,\mathrm{Orbits}\left(R\right)\right)$

$\left[24\,24\right]$

$\mathrm{Labels}\left(R\right)$

$\left[\mathrm{front}\,\mathrm{back}\,\mathrm{left}\,\mathrm{right}\,\mathrm{up}\,\mathrm{down}\right]$

$\mathrm{zip}\left(\mathrm{assign}\,\mathrm{Labels}\left(R\right)\,\mathrm{Generators}\left(R\right)\right)\:$

$\mathrm{SS}≔\mathrm{Subgroup}\left(\left\{{\mathrm{front}}^2·{\mathrm{back}}^2\,{\mathrm{right}}^2·{\mathrm{left}}^2\,{\mathrm{up}}^2·{\mathrm{down}}^2\right\}\,R\right)$

$\mathrm{SS}≔⟨\left(1\,8\right)\left(2\,7\right)\left(3\,6\right)\left(4\,5\right)\left(9\,25\right)\left(10\,26\right)\left(11\,27\right)\left(14\,30\right)\left(15\,31\right)\left(16\,32\right)\left(17\,33\right)\left(18\,34\right)\left(19\,35\right)\left(22\,38\right)\left(23\,39\right)\left(24\,40\right)\left(41\,48\right)\left(42\,47\right)\left(43\,46\right)\left(44\,45\right)\,\left(1\,48\right)\left(2\,47\right)\left(3\,46\right)\left(6\,43\right)\left(7\,42\right)\left(8\,41\right)\left(9\,32\right)\left(11\,30\right)\left(12\,29\right)\left(13\,28\right)\left(14\,27\right)\left(16\,25\right)\left(17\,24\right)\left(18\,23\right)\left(19\,22\right)\left(20\,21\right)\left(33\,40\right)\left(34\,39\right)\left(35\,38\right)\left(36\,37\right)\,\left(1\,41\right)\left(3\,43\right)\left(4\,44\right)\left(5\,45\right)\left(6\,46\right)\left(8\,48\right)\left(9\,16\right)\left(10\,15\right)\left(11\,14\right)\left(12\,13\right)\left(17\,40\right)\left(19\,38\right)\left(20\,37\right)\left(21\,36\right)\left(22\,35\right)\left(24\,33\right)\left(25\,32\right)\left(26\,31\right)\left(27\,30\right)\left(28\,29\right)⟩$

$\mathrm{IsAbelian}\left(\mathrm{SS}\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{SS}\right)$

$8$

The GroupTheory[RubiksCubeGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.