The symplectic group $Sp\left(n\,q\right)$ is the group of all $n\times n$ matrices over the field with $q$ elements that respect a fixed nondegenerate symplectic form. The integer $n$ must be even.

The SymplecticGroup( n, q ) command returns a permutation group isomorphic to the symplectic group $Sp\left(n\,q\right)$ .

Note that for $n=2$ the groups $Sp\left(n\,q\right)$ and $SL\left(n\,q\right)$ are isomorphic, so that a special linear group is returned in this case.

If either, or both, of n and q is non-numeric, then a symbolic group representing the symplectic group is returned.

The Sp( n, q ) command is provided as an abbreviation.

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

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

$G≔\mathrm{SymplecticGroup}\left(4\,5\right)$

$G≔\mathbf{Sp}\left(4\,5\right)$

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

${\left(2\right)}^7{\left(3\right)}^2{\left(5\right)}^4\left(13\right)$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(2\,G\right)\right)$

$128$

$\mathrm{S3}≔\mathrm{SylowSubgroup}\left(3\,G\right)$

$\mathrm{S3}≔⟨\left(1\,119\,168\right)\left(2\,160\,261\right)\left(3\,24\,272\right)\left(4\,253\,193\right)\left(5\,30\,205\right)\left(6\,305\,265\right)\left(7\,185\,127\right)\left(8\,46\,143\right)\left(9\,415\,197\right)\left(10\,245\,174\right)\left(11\,390\,510\right)\left(12\,34\,181\right)\left(13\,338\,134\right)\left(14\,342\,474\right)\left(15\,309\,444\right)\left(16\,220\,300\right)\left(17\,622\,406\right)\left(18\,249\,375\right)\left(19\,287\,410\right)\left(20\,621\,327\right)\left(21\,323\,240\right)\left(22\,623\,294\right)\left(23\,436\,437\right)\left(25\,487\,416\right)\left(27\,403\,333\right)\left(28\,624\,230\right)\left(29\,535\,536\right)\left(31\,547\,339\right)\left(33\,365\,366\right)\left(35\,558\,306\right)\left(37\,528\,549\right)\left(38\,552\,397\right)\left(39\,100\,277\right)\left(40\,44\,505\right)\left(41\,386\,244\right)\left(42\,120\,564\right)\left(43\,341\,496\right)\left(45\,470\,471\right)\left(47\,506\,246\right)\left(49\,458\,576\right)\left(50\,488\,316\right)\left(51\,155\,394\right)\left(52\,56\,557\right)\left(53\,450\,304\right)\left(54\,161\,578\right)\left(55\,248\,551\right)\left(57\,424\,493\right)\left(58\,497\,280\right)\left(59\,78\,210\right)\left(60\,64\,546\right)\left(61\,481\,337\right)\left(62\,186\,520\right)\left(63\,371\,572\right)\left(65\,446\,132\right)\left(66\,115\,494\right)\left(67\,94\,154\right)\left(68\,542\,519\right)\left(69\,554\,504\right)\left(70\,183\,178\right)\left(71\,99\,104\right)\left(72\,571\,545\right)\left(73\,267\,585\right)\left(74\,562\,236\right)\left(75\,184\,241\right)\left(76\,77\,146\right)\left(79\,550\,89\right)\left(80\,351\,511\right)\left(81\,503\,213\right)\left(82\,114\,313\right)\left(83\,87\,486\right)\left(84\,517\,414\right)\left(85\,254\,518\right)\left(86\,308\,603\right)\left(88\,382\,172\right)\left(90\,112\,113\right)\left(91\,480\,563\right)\left(92\,589\,556\right)\left(93\,117\,269\right)\left(95\,495\,484\right)\left(96\,200\,580\right)\left(97\,516\,225\right)\left(98\,118\,301\right)\left(101\,593\,145\right)\left(102\,377\,196\right)\left(103\,156\,570\right)\left(105\,449\,579\right)\left(106\,499\,575\right)\left(107\,251\,140\right)\left(108\,560\,453\right)\left(109\,176\,565\right)\left(110\,514\,331\right)\left(111\,252\,334\right)\left(116\,328\,303\right)\left(121\,608\,619\right)\left(122\,123\,137\right)\left(124\,322\,356\right)\left(126\,601\,399\right)\left(128\,538\,307\right)\left(129\,163\,164\right)\left(131\,490\,302\right)\left(133\,256\,257\right)\left(135\,472\,485\right)\left(136\,443\,247\right)\left(138\,521\,151\right)\left(139\,569\,553\right)\left(141\,457\,289\right)\left(142\,422\,476\right)\left(144\,372\,264\right)\left(147\,385\,605\right)\left(148\,574\,604\right)\left(149\,158\,202\right)\left(150\,512\,389\right)\left(152\,509\,292\right)\left(153\,159\,411\right)\left(157\,231\,413\right)\left(162\,615\,618\right)\left(165\,219\,429\right)\left(167\,612\,318\right)\left(169\,473\,417\right)\left(171\,508\,412\right)\left(173\,188\,189\right)\left(175\,373\,391\right)\left(177\,491\,588\right)\left(179\,350\,332\right)\left(180\,526\,378\right)\left(182\,407\,243\right)\left(187\,586\,616\right)\left(190\,402\,463\right)\left(192\,587\,282\right)\left(195\,561\,242\right)\left(198\,537\,525\right)\left(199\,374\,340\right)\left(201\,602\,498\right)\left(203\,527\,222\right)\left(204\,349\,421\right)\left(206\,440\,577\right)\left(207\,529\,597\right)\left(208\,568\,404\right)\left(209\,600\,435\right)\left(211\,357\,279\right)\left(212\,314\,464\right)\left(215\,260\,590\right)\left(216\,228\,355\right)\left(217\,293\,461\right)\left(218\,433\,347\right)\left(221\,428\,283\right)\left(223\,346\,582\right)\left(224\,380\,507\right)\left(226\,468\,383\right)\left(227\,352\,500\right)\left(229\,530\,320\right)\left(232\,344\,598\right)\left(233\,362\,381\right)\left(234\,425\,466\right)\left(235\,539\,555\right)\left(237\,534\,441\right)\left(238\,459\,432\right)\left(239\,270\,423\right)\left(250\,295\,336\right)\left(255\,599\,617\right)\left(258\,286\,532\right)\left(263\,513\,335\right)\left(266\,442\,454\right)\left(268\,548\,573\right)\left(271\,456\,348\right)\left(273\,478\,515\right)\left(274\,460\,610\right)\left(275\,606\,324\right)\left(276\,581\,455\right)\left(278\,430\,396\right)\left(284\,405\,354\right)\left(285\,501\,420\right)\left(288\,531\,400\right)\left(290\,311\,524\right)\left(291\,369\,559\right)\left(296\,419\,611\right)\left(297\,434\,370\right)\left(298\,492\,359\right)\left(299\,475\,523\right)\left(310\,426\,594\right)\left(312\,566\,364\right)\left(315\,395\,489\right)\left(319\,325\,462\right)\left(321\,360\,363\right)\left(326\,427\,401\right)\left(329\,439\,595\right)\left(330\,445\,502\right)\left(343\,584\,465\right)\left(345\,393\,522\right)\left(353\,614\,392\right)\left(358\,418\,607\right)\left(361\,467\,469\right)\left(367\,583\,431\right)\left(368\,596\,408\right)\left(376\,567\,409\right)\left(384\,592\,387\right)\left(438\,591\,533\right)\left(448\,609\,451\right)\left(479\,613\,482\right)\left(541\,620\,543\right)\,\left(1\,162\,372\right)\left(2\,255\,377\right)\left(3\,427\,468\right)\left(4\,187\,446\right)\left(5\,530\,362\right)\left(6\,213\,561\right)\left(7\,121\,382\right)\left(8\,461\,434\right)\left(9\,280\,490\right)\left(10\,316\,513\right)\left(11\,580\,225\right)\left(12\,354\,534\right)\left(13\,397\,508\right)\left(14\,521\,292\right)\left(15\,585\,236\right)\left(16\,51\,103\right)\left(17\,611\,139\right)\left(18\,565\,331\right)\left(19\,82\,66\right)\left(20\,596\,177\right)\left(21\,39\,145\right)\left(22\,598\,201\right)\left(23\,542\,507\right)\left(24\,401\,383\right)\left(25\,575\,157\right)\left(26\,194\,483\right)\left(27\,59\,89\right)\left(28\,595\,268\right)\left(29\,480\,559\right)\left(30\,320\,381\right)\left(31\,504\,250\right)\left(32\,130\,388\right)\left(33\,449\,515\right)\left(34\,284\,441\right)\left(35\,604\,116\right)\left(36\,262\,544\right)\left(37\,163\,312\right)\left(38\,412\,338\right)\left(40\,505\,44\right)\left(41\,592\,98\right)\left(42\,584\,560\right)\left(43\,288\,467\right)\left(45\,385\,577\right)\left(46\,217\,370\right)\left(47\,556\,182\right)\left(48\,170\,452\right)\left(49\,256\,209\right)\left(50\,335\,245\right)\left(52\,557\,56\right)\left(53\,609\,153\right)\left(54\,607\,571\right)\left(55\,325\,360\right)\left(57\,122\,393\right)\left(58\,302\,415\right)\left(60\,546\,64\right)\left(61\,613\,75\right)\left(62\,591\,512\right)\left(63\,221\,501\right)\left(65\,193\,616\right)\left(67\,421\,466\right)\left(68\,380\,437\right)\left(69\,295\,547\right)\left(70\,612\,199\right)\left(71\,271\,227\right)\left(72\,578\,418\right)\left(73\,74\,309\right)\left(76\,180\,238\right)\left(77\,526\,459\right)\left(78\,79\,403\right)\left(80\,188\,276\right)\left(81\,242\,305\right)\left(83\,486\,87\right)\left(84\,620\,111\right)\left(85\,583\,495\right)\left(86\,228\,433\right)\left(88\,127\,619\right)\left(90\,476\,359\right)\left(91\,369\,536\right)\left(92\,407\,506\right)\left(93\,590\,136\right)\left(94\,204\,234\right)\left(95\,518\,367\right)\left(96\,97\,390\right)\left(99\,456\,352\right)\left(100\,101\,323\right)\left(102\,261\,617\right)\left(104\,348\,500\right)\left(105\,478\,366\right)\left(106\,231\,487\right)\left(107\,601\,169\right)\left(108\,564\,343\right)\left(109\,110\,249\right)\left(112\,142\,298\right)\left(113\,422\,492\right)\left(114\,115\,287\right)\left(117\,215\,443\right)\left(118\,386\,387\right)\left(119\,615\,264\right)\left(120\,465\,453\right)\left(123\,522\,424\right)\left(124\,315\,289\right)\left(125\,214\,540\right)\left(126\,417\,140\right)\left(128\,149\,587\right)\left(129\,364\,549\right)\left(131\,197\,497\right)\left(132\,253\,586\right)\left(133\,435\,576\right)\left(134\,552\,171\right)\left(135\,324\,610\right)\left(137\,345\,493\right)\left(138\,509\,474\right)\left(141\,322\,395\right)\left(143\,293\,297\right)\left(144\,168\,618\right)\left(146\,378\,432\right)\left(147\,440\,471\right)\left(148\,328\,558\right)\left(150\,520\,438\right)\left(151\,152\,342\right)\left(154\,349\,425\right)\left(155\,156\,220\right)\left(158\,282\,538\right)\left(159\,450\,451\right)\left(160\,599\,196\right)\left(161\,358\,545\right)\left(164\,566\,528\right)\left(165\,212\,332\right)\left(166\,281\,477\right)\left(167\,340\,178\right)\left(172\,185\,608\right)\left(173\,455\,511\right)\left(174\,488\,263\right)\left(175\,223\,614\right)\left(176\,514\,375\right)\left(179\,219\,314\right)\left(181\,405\,237\right)\left(183\,318\,374\right)\left(184\,481\,482\right)\left(186\,533\,389\right)\left(189\,581\,351\right)\left(190\,396\,222\right)\left(191\,317\,447\right)\left(192\,307\,202\right)\left(195\,265\,503\right)\left(198\,404\,597\right)\left(200\,516\,510\right)\left(203\,402\,278\right)\left(205\,229\,233\right)\left(206\,470\,605\right)\left(207\,537\,208\right)\left(210\,550\,333\right)\left(211\,270\,286\right)\left(216\,218\,603\right)\left(224\,436\,519\right)\left(226\,272\,326\right)\left(230\,439\,573\right)\left(232\,602\,623\right)\left(235\,555\,539\right)\left(239\,258\,279\right)\left(240\,277\,593\right)\left(241\,337\,479\right)\left(243\,246\,589\right)\left(244\,384\,301\right)\left(247\,269\,260\right)\left(248\,462\,363\right)\left(251\,399\,473\right)\left(252\,517\,543\right)\left(254\,431\,484\right)\left(257\,600\,458\right)\left(259\,398\,379\right)\left(266\,290\,594\right)\left(267\,562\,444\right)\left(273\,365\,579\right)\left(274\,472\,275\right)\left(283\,285\,572\right)\left(291\,535\,563\right)\left(294\,344\,498\right)\left(296\,569\,622\right)\left(299\,523\,475\right)\left(300\,394\,570\right)\left(303\,306\,574\right)\left(304\,448\,411\right)\left(308\,355\,347\right)\left(310\,442\,311\right)\left(313\,494\,410\right)\left(319\,321\,551\right)\left(327\,368\,588\right)\left(329\,548\,624\right)\left(330\,502\,445\right)\left(334\,414\,541\right)\left(336\,339\,554\right)\left(341\,531\,469\right)\left(346\,392\,373\right)\left(350\,429\,464\right)\left(353\,391\,582\right)\left(356\,489\,457\right)\left(357\,423\,532\right)\left(361\,496\,400\right)\left(371\,428\,420\right)\left(376\,409\,567\right)\left(406\,419\,553\right)\left(408\,491\,621\right)\left(413\,416\,499\right)\left(426\,454\,524\right)\left(430\,527\,463\right)\left(460\,485\,606\right)\left(525\,568\,529\right)⟩$

$\mathrm{GroupOrder}\left(\mathrm{S3}\right)$

$9$

$\mathrm{IsCyclic}\left(\mathrm{S3}\right)$

$\mathrm{false}$

$\mathrm{IdentifySmallGroup}\left(\mathrm{S3}\right)$

$9,2$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(5\,G\right)\right)$

$625$

$\mathrm{IsTrivial}\left(\mathrm{PCore}\left(5\,G\right)\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(13\,G\right)\right)$

$13$

$G≔\mathrm{SymplecticGroup}\left(4\,3\right)$

$G≔\mathbf{Sp}\left(4\,3\right)$

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

$80$

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

$\mathrm{false}$

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

$2$

$\mathrm{SymplecticGroup}\left(2\,5\right)$

$\mathbf{SL}\left(2\,5\right)$

$\mathrm{AreIsomorphic}\left(\mathrm{SymplecticGroup}\left(4\,2\right)\,\mathrm{Symm}\left(6\right)\right)$

$\mathrm{true}$

$G≔\mathrm{SymplecticGroup}\left(6\,q\right)$

$G≔\mathbf{Sp}\left(6\,q\right)$

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

$q^9\left(q^2-1\right)\left(q^4-1\right)\left(q^6-1\right)$

$\mathrm{ClassNumber}\left(\mathrm{SymplecticGroup}\left(8\,q\right)\right)$

$\left\{\begin{array}{cc}5q+\left(q+1\right)q+4q^2+\left(q^2+q+3\right)q+q^4+q^3+7& q::\mathrm{even}\\ 25q+51+\left(q+4\right)q+11q^2+\left(q^2+4q+10\right)q+q^4+4q^3& \mathrm{otherwise}\end{array}\right.$

$\mathrm{ClassNumber}\left(\mathrm{SymplecticGroup}\left(4\,{11}^k\right)\right)assumingk::\mathrm{posint}$

$5{11}^k+10+{\left({11}^k\right)}^2$

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

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

The GroupTheory[SymplecticGroup] command was updated in Maple 2020.