The Orbit( alpha, G ) command returns the orbit of the point alpha under the action of the permutation group G.

The returned value is an object that supports the following methods.

The Orbits( G ) command returns the set of all orbits of the permutation group G.

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

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

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

$\mathrm{O1}≔\mathrm{Orbit}\left(1\,G\right)$

$\mathrm{O1}≔1^{⟨\left(6\,25\,43\,16\right)\left(7\,28\,42\,13\right)\left(8\,30\,41\,11\right)\left(17\,19\,24\,22\right)\left(18\,21\,23\,20\right)\,\left(1\,14\,48\,27\right)\left(2\,12\,47\,29\right)\left(3\,9\,46\,32\right)\left(33\,35\,40\,38\right)\left(34\,37\,39\,36\right)\,\left(1\,17\,41\,40\right)\left(4\,20\,44\,37\right)\left(6\,22\,46\,35\right)\left(9\,11\,16\,14\right)\left(10\,13\,15\,12\right)\,\left(3\,38\,43\,19\right)\left(5\,36\,45\,21\right)\left(8\,33\,48\,24\right)\left(25\,27\,32\,30\right)\left(26\,29\,31\,28\right)\,\left(1\,3\,8\,6\right)\left(2\,5\,7\,4\right)\left(9\,33\,25\,17\right)\left(10\,34\,26\,18\right)\left(11\,35\,27\,19\right)\,\left(14\,22\,30\,38\right)\left(15\,23\,31\,39\right)\left(16\,24\,32\,40\right)\left(41\,43\,48\,46\right)\left(42\,45\,47\,44\right)⟩}$

$\mathrm{numelems}\left(\mathrm{O1}\right)$

$24$

$2\mathbf{in}\mathrm{O1}$

$\mathrm{false}$

$\mathrm{O2}≔\mathrm{Orbit}\left(2\,G\right)$

$\mathrm{O2}≔2^{⟨\left(6\,25\,43\,16\right)\left(7\,28\,42\,13\right)\left(8\,30\,41\,11\right)\left(17\,19\,24\,22\right)\left(18\,21\,23\,20\right)\,\left(1\,14\,48\,27\right)\left(2\,12\,47\,29\right)\left(3\,9\,46\,32\right)\left(33\,35\,40\,38\right)\left(34\,37\,39\,36\right)\,\left(1\,17\,41\,40\right)\left(4\,20\,44\,37\right)\left(6\,22\,46\,35\right)\left(9\,11\,16\,14\right)\left(10\,13\,15\,12\right)\,\left(3\,38\,43\,19\right)\left(5\,36\,45\,21\right)\left(8\,33\,48\,24\right)\left(25\,27\,32\,30\right)\left(26\,29\,31\,28\right)\,\left(1\,3\,8\,6\right)\left(2\,5\,7\,4\right)\left(9\,33\,25\,17\right)\left(10\,34\,26\,18\right)\left(11\,35\,27\,19\right)\,\left(14\,22\,30\,38\right)\left(15\,23\,31\,39\right)\left(16\,24\,32\,40\right)\left(41\,43\,48\,46\right)\left(42\,45\,47\,44\right)⟩}$

$\mathrm{numelems}\left(\mathrm{O2}\right)$

$24$

$\mathrm{orbs}≔\mathrm{Orbits}\left(G\right)$

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

$\mathrm{nops}\left(\mathrm{orbs}\right)$

$2$

$\mathrm{Elements}\left(\mathrm{O2}\right)$

$\left\{2\,4\,5\,7\,10\,12\,13\,15\,18\,20\,21\,23\,26\,28\,29\,31\,34\,36\,37\,39\,42\,44\,45\,47\right\}$

The GroupTheory[Orbit] and GroupTheory[Orbits] commands were introduced in Maple 17.

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