The RandomSmallGroup() command returns a randomly selected group from the database of small groups, as a permutation group.

The id option controls how the group is selected. If the option id = true (or just id) is passed, then a randomly selected order in the range 1 .. 511 is first selected, and then, within the groups of that order, a random group is returned. If the id = false option is passed, then a truly (pseudo-)randomly selected group is returned from the database of small groups.  Note that most groups in the small groups database have order equal to $256$, so this is usually not what is wanted, which is why the default option is id = true.

The form option determines the form of what is returned. By default, a permutation group is returned. To have a finitely presented group returned, use the form = "fpgroup" option. Sometimes, only a valid small group ID is required, in which case, use the form = "id" option.

$''SEED\; ='',59308905600$

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

$\mathrm{RandomSmallGroup}\left(\right)$

$⟨\left(1\,2\right)\left(3\,8\right)\left(4\,7\right)\left(5\,10\right)\left(6\,9\right)\left(11\,22\right)\left(12\,24\right)\left(13\,23\right)\left(14\,19\right)\left(15\,21\right)\left(16\,20\right)\left(17\,26\right)\left(18\,25\right)\left(27\,42\right)\left(28\,41\right)\left(29\,44\right)\left(30\,43\right)\left(31\,38\right)\left(32\,37\right)\left(33\,40\right)\left(34\,39\right)\left(35\,46\right)\left(36\,45\right)\left(47\,62\right)\left(48\,61\right)\left(49\,64\right)\left(50\,63\right)\left(51\,58\right)\left(52\,57\right)\left(53\,60\right)\left(54\,59\right)\left(55\,66\right)\left(56\,65\right)\left(67\,82\right)\left(68\,81\right)\left(69\,84\right)\left(70\,83\right)\left(71\,78\right)\left(72\,77\right)\left(73\,80\right)\left(74\,79\right)\left(75\,86\right)\left(76\,85\right)\left(87\,102\right)\left(88\,101\right)\left(89\,104\right)\left(90\,103\right)\left(91\,98\right)\left(92\,97\right)\left(93\,100\right)\left(94\,99\right)\left(95\,106\right)\left(96\,105\right)\left(107\,122\right)\left(108\,121\right)\left(109\,124\right)\left(110\,123\right)\left(111\,118\right)\left(112\,117\right)\left(113\,120\right)\left(114\,119\right)\left(115\,126\right)\left(116\,125\right)\left(127\,142\right)\left(128\,141\right)\left(129\,144\right)\left(130\,143\right)\left(131\,138\right)\left(132\,137\right)\left(133\,140\right)\left(134\,139\right)\left(135\,146\right)\left(136\,145\right)\left(147\,162\right)\left(148\,161\right)\left(149\,164\right)\left(150\,163\right)\left(151\,158\right)\left(152\,157\right)\left(153\,160\right)\left(154\,159\right)\left(155\,166\right)\left(156\,165\right)\left(167\,180\right)\left(168\,179\right)\left(169\,182\right)\left(170\,181\right)\left(171\,176\right)\left(172\,175\right)\left(173\,178\right)\left(174\,177\right)\left(183\,190\right)\left(184\,189\right)\left(185\,188\right)\left(186\,187\right)\,\left(1\,3\,11\,14\,4\right)\left(2\,7\,19\,22\,8\right)\left(5\,12\,27\,31\,15\right)\left(6\,13\,28\,32\,16\right)\left(9\,20\,37\,41\,23\right)\left(10\,21\,38\,42\,24\right)\left(17\,29\,47\,51\,33\right)\left(18\,30\,48\,52\,34\right)\left(25\,39\,57\,61\,43\right)\left(26\,40\,58\,62\,44\right)\left(35\,49\,67\,71\,53\right)\left(36\,50\,68\,72\,54\right)\left(45\,59\,77\,81\,63\right)\left(46\,60\,78\,82\,64\right)\left(55\,69\,87\,91\,73\right)\left(56\,70\,88\,92\,74\right)\left(65\,79\,97\,101\,83\right)\left(66\,80\,98\,102\,84\right)\left(75\,89\,107\,111\,93\right)\left(76\,90\,108\,112\,94\right)\left(85\,99\,117\,121\,103\right)\left(86\,100\,118\,122\,104\right)\left(95\,109\,127\,131\,113\right)\left(96\,110\,128\,132\,114\right)\left(105\,119\,137\,141\,123\right)\left(106\,120\,138\,142\,124\right)\left(115\,129\,147\,151\,133\right)\left(116\,130\,148\,152\,134\right)\left(125\,139\,157\,161\,143\right)\left(126\,140\,158\,162\,144\right)\left(135\,149\,167\,171\,153\right)\left(136\,150\,168\,172\,154\right)\left(145\,159\,175\,179\,163\right)\left(146\,160\,176\,180\,164\right)\left(155\,169\,183\,185\,173\right)\left(156\,170\,184\,186\,174\right)\left(165\,177\,187\,189\,181\right)\left(166\,178\,188\,190\,182\right)\,\left(1\,5\,17\,35\,55\,75\,95\,115\,135\,155\,156\,136\,116\,96\,76\,56\,36\,18\,6\right)\left(2\,9\,25\,45\,65\,85\,105\,125\,145\,165\,166\,146\,126\,106\,86\,66\,46\,26\,10\right)\left(3\,12\,29\,49\,69\,89\,109\,129\,149\,169\,170\,150\,130\,110\,90\,70\,50\,30\,13\right)\left(4\,15\,33\,53\,73\,93\,113\,133\,153\,173\,174\,154\,134\,114\,94\,74\,54\,34\,16\right)\left(7\,20\,39\,59\,79\,99\,119\,139\,159\,177\,178\,160\,140\,120\,100\,80\,60\,40\,21\right)\left(8\,23\,43\,63\,83\,103\,123\,143\,163\,181\,182\,164\,144\,124\,104\,84\,64\,44\,24\right)\left(11\,27\,47\,67\,87\,107\,127\,147\,167\,183\,184\,168\,148\,128\,108\,88\,68\,48\,28\right)\left(14\,31\,51\,71\,91\,111\,131\,151\,171\,185\,186\,172\,152\,132\,112\,92\,72\,52\,32\right)\left(19\,37\,57\,77\,97\,117\,137\,157\,175\,187\,188\,176\,158\,138\,118\,98\,78\,58\,38\right)\left(22\,41\,61\,81\,101\,121\,141\,161\,179\,189\,190\,180\,162\,142\,122\,102\,82\,62\,42\right)⟩$

$\mathrm{RandomSmallGroup}\left('\mathrm{id}'=\mathrm{false}\right)$

$\mathrm{\⟨a\; permutation\; group\; on\; 256\; letters\; with\; 8\; generators\⟩}$

$G≔\mathrm{RandomSmallGroup}\left('\mathrm{id}'\,'\mathrm{maxorder}'=200\right)$

$G≔⟨\left(1\,2\,4\,6\,8\,10\,12\,14\,16\,18\,20\,22\,24\,26\,28\,30\,32\,34\,36\,38\,40\,42\,44\,46\,48\,50\,52\,54\,56\,58\,60\,62\,64\,66\,68\,70\,72\,74\,76\,78\,80\,82\,84\,86\,88\,90\,92\,94\,96\,98\,100\,102\,104\,106\,108\,110\,112\,114\,116\,118\,120\,122\,124\,126\,128\,130\,132\,134\,136\,138\,140\,142\,144\,146\,148\,150\,152\,154\,156\,158\,160\,162\,164\,166\,168\,170\,172\,173\,171\,169\,167\,165\,163\,161\,159\,157\,155\,153\,151\,149\,147\,145\,143\,141\,139\,137\,135\,133\,131\,129\,127\,125\,123\,121\,119\,117\,115\,113\,111\,109\,107\,105\,103\,101\,99\,97\,95\,93\,91\,89\,87\,85\,83\,81\,79\,77\,75\,73\,71\,69\,67\,65\,63\,61\,59\,57\,55\,53\,51\,49\,47\,45\,43\,41\,39\,37\,35\,33\,31\,29\,27\,25\,23\,21\,19\,17\,15\,13\,11\,9\,7\,5\,3\right)⟩$

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

$173$

$G≔\mathrm{RandomSmallGroup}\left('\mathrm{form}'=''fpgroup''\right)$

$G≔⟨\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\,\mathrm{a4}\mid {\mathrm{a1}}^2\,{\mathrm{a2}}^2\,{\mathrm{a3}}^2\,{\mathrm{a1}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a1}\mathrm{a2}\,{\mathrm{a1}}^{-1}{\mathrm{a3}}^{-1}\mathrm{a1}\mathrm{a3}\,{\mathrm{a1}}^{-1}{\mathrm{a4}}^{-1}\mathrm{a1}\mathrm{a4}\,{\mathrm{a2}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a2}\mathrm{a1}\,{\mathrm{a2}}^{-1}{\mathrm{a3}}^{-1}\mathrm{a2}\mathrm{a3}\,{\mathrm{a2}}^{-1}{\mathrm{a4}}^{-1}\mathrm{a2}\mathrm{a4}\,{\mathrm{a3}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a3}\mathrm{a1}\,{\mathrm{a3}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a3}\mathrm{a2}\,{\mathrm{a3}}^{-1}{\mathrm{a4}}^{-1}\mathrm{a3}\mathrm{a4}\,{\mathrm{a4}}^{-1}{\mathrm{a1}}^{-1}\mathrm{a4}\mathrm{a1}\,{\mathrm{a4}}^{-1}{\mathrm{a2}}^{-1}\mathrm{a4}\mathrm{a2}\,{\mathrm{a4}}^{-1}{\mathrm{a3}}^{-1}\mathrm{a4}\mathrm{a3}\,{\mathrm{a4}}^{37}⟩$

$G≔\mathrm{RandomSmallGroup}\left('\mathrm{maxorder}'=100\,'\mathrm{form}'=''fpgroup''\right)$

$G≔⟨\mathrm{\_a}\,\mathrm{\_b}\mid {\mathrm{\_b}}^3\,{\mathrm{\_a}}^{-1}{\mathrm{\_b}}^{-1}\mathrm{\_a}\mathrm{\_b}\,{\mathrm{\_a}}^9⟩$

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

$27$

$\mathrm{RandomSmallGroup}\left('\mathrm{form}'=''id''\right)$

$\left[153\,2\right]$

$\mathrm{id}≔\mathrm{RandomSmallGroup}\left('\mathrm{form}'='\mathrm{id}'\,'\mathrm{maxorder}'=150\right)$

$\mathrm{id}≔\left[97\,1\right]$

$\mathrm{evalb}\left(\mathrm{id}\left[1\right]\le 150\right)$

$\mathrm{true}$

The GroupTheory[RandomSmallGroup] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.