The Steinberg group ${}^3\mathrm{D}_4\left(q\right)$ , for a prime power $q$, is a simple group of Lie type.

The Steinberg3D4( q ) command returns a symbolic group representing the Steinberg group ${}^3\mathrm{D}_4\left(q\right)$ or, if $q=2$ or $q=3$, a permutation group isomorphic to ${}^3\mathrm{D}_4\left(q\right)$ .

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

$G≔\mathrm{Steinberg3D4}\left(2\right)$

$G≔⟨\left(1\,2\right)\left(3\,5\right)\left(4\,7\right)\left(6\,10\right)\left(8\,13\right)\left(9\,15\right)\left(11\,18\right)\left(12\,20\right)\left(14\,23\right)\left(16\,26\right)\left(17\,28\right)\left(19\,31\right)\left(21\,34\right)\left(22\,35\right)\left(24\,38\right)\left(25\,33\right)\left(27\,42\right)\left(29\,45\right)\left(30\,47\right)\left(32\,50\right)\left(36\,55\right)\left(37\,57\right)\left(39\,60\right)\left(40\,62\right)\left(41\,64\right)\left(43\,67\right)\left(44\,54\right)\left(46\,71\right)\left(48\,74\right)\left(51\,78\right)\left(52\,80\right)\left(53\,82\right)\left(56\,86\right)\left(58\,89\right)\left(59\,91\right)\left(61\,94\right)\left(63\,96\right)\left(65\,99\right)\left(66\,98\right)\left(68\,103\right)\left(69\,104\right)\left(70\,106\right)\left(72\,109\right)\left(73\,111\right)\left(75\,114\right)\left(76\,116\right)\left(77\,118\right)\left(79\,121\right)\left(81\,124\right)\left(83\,127\right)\left(84\,129\right)\left(85\,131\right)\left(88\,135\right)\left(90\,138\right)\left(92\,141\right)\left(93\,143\right)\left(95\,146\right)\left(97\,148\right)\left(100\,152\right)\left(101\,154\right)\left(102\,156\right)\left(105\,159\right)\left(107\,162\right)\left(108\,164\right)\left(110\,165\right)\left(112\,167\right)\left(113\,169\right)\left(115\,171\right)\left(117\,174\right)\left(119\,177\right)\left(120\,179\right)\left(122\,182\right)\left(123\,183\right)\left(125\,186\right)\left(126\,188\right)\left(128\,190\right)\left(130\,193\right)\left(132\,196\right)\left(133\,198\right)\left(134\,200\right)\left(136\,203\right)\left(137\,205\right)\left(139\,207\right)\left(140\,209\right)\left(142\,212\right)\left(144\,215\right)\left(145\,217\right)\left(147\,220\right)\left(149\,222\right)\left(150\,224\right)\left(151\,226\right)\left(153\,229\right)\left(155\,232\right)\left(157\,235\right)\left(158\,236\right)\left(160\,239\right)\left(161\,241\right)\left(163\,244\right)\left(166\,247\right)\left(168\,250\right)\left(170\,253\right)\left(172\,255\right)\left(173\,257\right)\left(175\,260\right)\left(176\,262\right)\left(178\,265\right)\left(181\,269\right)\left(184\,273\right)\left(185\,275\right)\left(187\,277\right)\left(191\,281\right)\left(192\,283\right)\left(195\,242\right)\left(197\,288\right)\left(199\,290\right)\left(201\,293\right)\left(202\,295\right)\left(204\,297\right)\left(206\,300\right)\left(208\,303\right)\left(210\,305\right)\left(211\,307\right)\left(213\,310\right)\left(214\,311\right)\left(216\,274\right)\left(218\,316\right)\left(219\,318\right)\left(221\,321\right)\left(223\,285\right)\left(225\,325\right)\left(227\,328\right)\left(228\,330\right)\left(230\,280\right)\left(231\,334\right)\left(233\,336\right)\left(234\,338\right)\left(237\,339\right)\left(238\,341\right)\left(240\,344\right)\left(243\,346\right)\left(245\,349\right)\left(246\,351\right)\left(248\,354\right)\left(249\,356\right)\left(251\,359\right)\left(252\,361\right)\left(254\,364\right)\left(256\,367\right)\left(258\,263\right)\left(259\,371\right)\left(261\,373\right)\left(264\,376\right)\left(266\,317\right)\left(267\,379\right)\left(268\,375\right)\left(270\,383\right)\left(271\,385\right)\left(272\,387\right)\left(276\,391\right)\left(278\,358\right)\left(279\,393\right)\left(282\,397\right)\left(284\,399\right)\left(286\,401\right)\left(287\,403\right)\left(289\,406\right)\left(291\,408\right)\left(292\,410\right)\left(294\,413\right)\left(296\,416\right)\left(298\,419\right)\left(299\,421\right)\left(301\,340\right)\left(304\,426\right)\left(306\,429\right)\left(308\,312\right)\left(309\,432\right)\left(313\,436\right)\left(314\,437\right)\left(315\,439\right)\left(319\,444\right)\left(320\,446\right)\left(322\,448\right)\left(324\,450\right)\left(326\,402\right)\left(327\,452\right)\left(329\,455\right)\left(331\,457\right)\left(335\,462\right)\left(337\,463\right)\left(342\,465\right)\left(343\,467\right)\left(345\,380\right)\left(347\,352\right)\left(348\,471\right)\left(350\,474\right)\left(353\,476\right)\left(355\,478\right)\left(357\,480\right)\left(360\,422\right)\left(362\,386\right)\left(363\,484\right)\left(365\,425\right)\left(366\,423\right)\left(368\,487\right)\left(369\,488\right)\left(372\,491\right)\left(374\,492\right)\left(377\,495\right)\left(378\,496\right)\left(381\,497\right)\left(382\,499\right)\left(384\,502\right)\left(388\,506\right)\left(389\,508\right)\left(390\,510\right)\left(392\,513\right)\left(394\,516\right)\left(395\,459\right)\left(398\,518\right)\left(400\,521\right)\left(404\,524\right)\left(405\,525\right)\left(407\,515\right)\left(409\,529\right)\left(411\,532\right)\left(412\,534\right)\left(414\,536\right)\left(415\,537\right)\left(417\,540\right)\left(418\,539\right)\left(420\,544\right)\left(424\,546\right)\left(427\,549\right)\left(428\,551\right)\left(430\,553\right)\left(431\,550\right)\left(433\,557\right)\left(434\,559\right)\left(435\,561\right)\left(438\,565\right)\left(440\,568\right)\left(441\,569\right)\left(442\,571\right)\left(443\,573\right)\left(445\,576\right)\left(447\,578\right)\left(449\,579\right)\left(451\,581\right)\left(453\,584\right)\left(454\,586\right)\left(456\,589\right)\left(458\,591\right)\left(460\,594\right)\left(461\,545\right)\left(464\,596\right)\left(468\,585\right)\left(469\,483\right)\left(470\,601\right)\left(472\,602\right)\left(473\,604\right)\left(475\,606\right)\left(477\,608\right)\left(479\,609\right)\left(481\,612\right)\left(482\,614\right)\left(486\,616\right)\left(489\,595\right)\left(490\,619\right)\left(493\,621\right)\left(494\,519\right)\left(498\,625\right)\left(500\,620\right)\left(501\,628\right)\left(504\,629\right)\left(505\,631\right)\left(507\,632\right)\left(509\,634\right)\left(511\,600\right)\left(512\,638\right)\left(514\,641\right)\left(517\,535\right)\left(520\,592\right)\left(522\,647\right)\left(523\,640\right)\left(526\,651\right)\left(527\,563\right)\left(528\,654\right)\left(530\,657\right)\left(531\,603\right)\left(533\,643\right)\left(538\,661\right)\left(541\,659\right)\left(542\,653\right)\left(547\,574\right)\left(548\,635\right)\left(552\,618\right)\left(554\,665\right)\left(555\,673\right)\left(556\,660\right)\left(558\,676\right)\left(560\,679\right)\left(562\,682\right)\left(564\,633\right)\left(566\,658\right)\left(567\,613\right)\left(570\,688\right)\left(572\,666\right)\left(575\,693\right)\left(577\,694\right)\left(580\,697\right)\left(582\,698\right)\left(583\,639\right)\left(587\,683\right)\left(590\,704\right)\left(593\,663\right)\left(597\,691\right)\left(598\,702\right)\left(599\,709\right)\left(605\,715\right)\left(607\,717\right)\left(610\,687\right)\left(611\,720\right)\left(615\,723\right)\left(617\,669\right)\left(622\,727\right)\left(624\,728\right)\left(626\,675\right)\left(627\,730\right)\left(630\,732\right)\left(636\,648\right)\left(637\,736\right)\left(642\,664\right)\left(644\,655\right)\left(645\,722\right)\left(646\,672\right)\left(649\,745\right)\left(650\,746\right)\left(652\,716\right)\left(656\,749\right)\left(662\,701\right)\left(667\,708\right)\left(668\,754\right)\left(670\,707\right)\left(671\,695\right)\left(674\,719\right)\left(677\,755\right)\left(678\,737\right)\left(680\,726\right)\left(684\,706\right)\left(685\,765\right)\left(686\,767\right)\left(689\,770\right)\left(692\,773\right)\left(696\,775\right)\left(699\,712\right)\left(700\,739\right)\left(703\,778\right)\left(705\,763\right)\left(710\,782\right)\left(711\,783\right)\left(713\,784\right)\left(714\,785\right)\left(718\,786\right)\left(721\,789\right)\left(724\,791\right)\left(725\,757\right)\left(729\,793\right)\left(731\,794\right)\left(733\,760\right)\left(734\,795\right)\left(735\,787\right)\left(738\,798\right)\left(740\,799\right)\left(741\,801\right)\left(742\,753\right)\left(743\,744\right)\left(747\,796\right)\left(748\,756\right)\left(750\,772\right)\left(751\,804\right)\left(752\,805\right)\left(758\,780\right)\left(759\,777\right)\left(761\,766\right)\left(762\,807\right)\left(764\,790\right)\left(768\,808\right)\left(769\,809\right)\left(771\,810\right)\left(774\,797\right)\left(776\,800\right)\left(779\,788\right)\left(781\,812\right)\left(792\,816\right)\left(802\,817\right)\left(803\,811\right)\left(806\,815\right)\left(813\,818\right)\left(814\,819\right)\,\left(1\,3\,6\,11\,19\,32\,51\,79\,122\right)\left(2\,4\,8\,14\,24\,39\,61\,95\,147\right)\left(5\,9\,16\,27\,43\,68\,67\,102\,157\right)\left(7\,12\,21\right)\left(10\,17\,29\,46\,72\,110\,166\,248\,355\right)\left(13\,22\,36\,56\,87\,134\,201\,294\,414\right)\left(15\,25\,40\,63\,97\,149\,223\,324\,109\right)\left(18\,30\,48\,75\,115\,172\,256\,368\,446\right)\left(20\,33\,52\,81\,125\,187\,278\,89\,137\right)\left(23\,37\,58\,90\,139\,208\,179\,267\,380\right)\left(26\,41\,65\,100\,153\,230\,333\,460\,595\right)\left(28\,44\,69\,105\,160\,240\,260\,71\,108\right)\left(31\,49\,76\,117\,175\,261\,361\,482\,62\right)\left(34\,53\,83\,128\,191\,282\,257\,369\,489\right)\left(35\,54\,84\,130\,194\,285\,182\,271\,386\right)\left(38\,59\,92\,142\,213\,167\,249\,357\,354\right)\left(42\,66\,101\,155\,233\,337\,106\,161\,242\right)\left(45\,70\,107\,163\,245\,350\,475\,255\,366\right)\left(47\,73\,112\,168\,251\,360\,481\,613\,722\right)\left(50\,77\,119\,178\,266\,159\,238\,342\,466\right)\left(55\,85\,132\,197\,224\,156\,234\,300\,423\right)\left(57\,88\,136\,204\,298\,420\,437\,564\,684\right)\left(60\,93\,144\,216\,314\,438\,566\,686\,768\right)\left(64\,98\,150\,225\,326\,451\,582\,604\,714\right)\left(74\,113\,138\,206\,301\,424\,547\,608\,719\right)\left(78\,120\,180\,268\,381\,498\,626\,359\,462\right)\left(80\,123\,184\,274\,389\,509\,635\,253\,363\right)\left(82\,126\,189\,280\,395\,517\,643\,325\,235\right)\left(86\,133\,199\,291\,409\,530\,183\,272\,310\right)\left(91\,140\,210\,306\,269\,382\,500\,413\,448\right)\left(94\,145\,218\,317\,442\,572\,307\,430\,554\right)\left(96\,121\,181\,270\,384\,503\,399\,520\,645\right)\left(99\,151\,227\,329\,200\,292\,411\,533\,103\right)\left(104\,158\,237\,340\,464\,597\,124\,185\,276\right)\left(111\,131\,195\,286\,402\,523\,648\,744\,732\right)\left(114\,170\,164\,246\,352\,471\,452\,583\,699\right)\left(116\,173\,258\,370\,379\,196\,287\,404\,502\right)\left(118\,176\,263\,375\,346\,273\,388\,507\,236\right)\left(127\,177\,264\,377\,406\,527\,653\,647\,743\right)\left(129\,192\,174\,259\,372\,436\,563\,683\,764\right)\left(135\,202\,229\,332\,459\,593\,706\,478\,165\right)\left(141\,211\,308\,431\,555\,674\,760\,783\,673\right)\left(143\,214\,312\,435\,562\,467\,599\,710\,186\right)\left(146\,219\,319\,445\,577\,621\,344\,468\,600\right)\left(148\,221\,322\,449\,580\,651\,364\,485\,303\right)\left(152\,228\,331\,458\,592\,499\,627\,731\,717\right)\left(154\,231\,335\,339\,367\,486\,617\,497\,624\right)\left(162\,243\,347\,470\,429\,450\,484\,615\,724\right)\left(169\,252\,362\,483\,349\,473\,605\,581\,474\right)\left(171\,254\,365\,262\,374\,493\,622\,586\,701\right)\left(188\,279\,394\,283\,398\,519\,594\,707\,629\right)\left(190\,193\,284\,400\,522\,232\,207\,302\,425\right)\left(198\,289\,239\,343\,265\,378\,492\,222\,323\right)\left(203\,296\,417\,541\,664\,755\,328\,454\,587\right)\left(205\,299\,422\,457\,455\,588\,702\,463\,373\right)\left(209\,304\,427\,550\,250\,358\,247\,353\,477\right)\left(212\,309\,433\,558\,677\,602\,712\,606\,716\right)\left(215\,313\,341\,290\,407\,496\,488\,618\,403\right)\left(217\,315\,440\,444\,575\,679\,534\,659\,752\right)\left(220\,320\,351\,387\,505\,416\,539\,662\,754\right)\left(226\,327\,453\,585\,700\,777\,808\,579\,696\right)\left(241\,345\,469\,601\,711\,356\,479\,610\,576\right)\left(244\,348\,472\,603\,713\,775\,786\,549\,669\right)\left(275\,390\,511\,637\,737\,778\,559\,678\,762\right)\left(277\,392\,514\,642\,741\,297\,418\,542\,665\right)\left(281\,396\,397\,518\,644\,536\,421\,545\,516\right)\left(288\,405\,526\,652\,748\,791\,694\,321\,447\right)\left(293\,412\,535\,660\,753\,439\,567\,383\,501\right)\left(295\,415\,538\,540\,663\,619\,725\,792\,789\right)\left(305\,428\,552\,671\,759\,785\,401\,338\,336\right)\left(311\,434\,560\,680\,763\,318\,443\,574\,692\right)\left(316\,441\,570\,689\,771\,784\,814\,697\,776\right)\left(330\,456\,590\,596\,708\,730\,782\,813\,804\right)\left(334\,461\,408\,528\,655\,578\,695\,510\,636\right)\left(371\,490\,620\,726\,521\,646\,742\,802\,584\right)\left(376\,494\,623\,682\,727\,591\,705\,780\,811\right)\left(385\,504\,630\,733\,551\,670\,614\,723\,487\right)\left(391\,512\,639\,532\,557\,675\,513\,640\,739\right)\left(393\,515\,569\,657\,750\,628\,632\,571\,690\right)\left(410\,531\,658\,751\,537\,508\,633\,491\,506\right)\left(419\,543\,666\,756\,616\,561\,681\,544\,661\right)\left(426\,548\,668\,758\,806\,654\,495\,465\,598\right)\left(432\,556\,480\,611\,546\,667\,757\,529\,656\right)\left(476\,607\,718\,787\,815\,816\,568\,687\,769\right)\left(524\,649\,612\,721\,790\,638\,738\,794\,773\right)\left(525\,650\,747\,704\,779\,693\,774\,688\,728\right)\left(553\,672\,609\,573\,691\,772\,745\,589\,703\right)\left(565\,685\,766\,805\,736\,797\,749\,631\,734\right)\left(625\,729\,676\,761\,799\,746\,803\,709\,781\right)\left(634\,735\,796\,807\,818\,810\,765\,801\,795\right)\left(641\,740\,800\,817\,798\,812\,770\,767\,715\right)\left(720\,788\,809\right)⟩$

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

$211341312$

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

$819$

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

$\mathrm{true}$

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

$\mathrm{true}$

$G≔\mathrm{Steinberg3D4}\left(3\right)\:$

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

$26572$

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

$\mathrm{true}$

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

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{Steinberg3D4}\left(4096\right)\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{Steinberg3D4}\left(27\right)\right)$

$11956114445971661401099296184508431605312$

$\mathrm{ClassNumber}\left(\mathrm{Steinberg3D4}\left(32\right)\right)$

$1082405$

$G≔\mathrm{Steinberg3D4}\left(q\right)$

$G≔{}^{3}D_4\left(q\right)$

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

$q^4+q^3+q^2+q+5+\mathrm{irem}\left(q\,2\right)$

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

$\mathrm{true}$

The GroupTheory[Steinberg3D4] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.