The IdentifyFrobeniusGroup command takes an option of the form 'assign' = iso, where iso is an unassigned name.

The AllFrobeniusGroups command takes an option of the form 'output' = "list" or "iterator".

The Frobenius groups database contains all the Frobenius groups of order less than or equal to $15000$, with certain exceptions, namely, those orders for which the Frobenius kernel has order divisible by $1024$. The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders, a complete and irredundant list of isomorphism type representatives of groups is given. These groups are available as permutation groups, and their Frobenius kernels and Frobenius complements are pre-computed.

The exceptional orders $n$ for which Frobenius groups of order $n$ have not yet been fully computed, and are therefore not complete in the database are: $3072$ and $12288$. For these orders, only those Frobenius groups for which the Frobenius kernel is Abelian are included in the database.

The FrobeniusGroup( n, d ) command returns the $d$-th Frobenius group of order $n$ in the Frobenius groups database.  The value of $n$ must be at most 15000 and not among those exceptions listed above.  The value of $d$ must be less than or equal to the number of Frobenius groups of order $n$. The syntax FrobeniusGroup( [ n, d ] ) is also accepted.

The NumFrobeniusGroups( n ) command returns the number of Frobenius groups of order $n$, where $n$ is an integer for which the Frobenius groups of order $n$ are known to Maple. A value of $0$ is returned if it is known that there are no Frobenius groups of order $n$.  If the Frobenius groups of order $n$ are known, then a positive integral value is returned.  Otherwise, an exception is raised.

The IdentifyFrobeniusGroup( G ) command attempts to locate an isomorphic copy of the given Frobenius group G in the database of Frobenius groups. If G is isomorphic to the $d$-th Frobenius group of order $n$ in the database, then the pair (n, d) is returned. If G is not a Frobenius group, or if the Frobenius groups of the same order as the order of G are not known (in the database), then an exception is raised.

An option of the form 'assign' = iso causes IdentifyFrobeniusGroup to compute an isomorphism from the input Frobenius group to the Frobenius group in the database. If an exception occurs in the presence of this option, then the name iso is not assigned a value.

The algorithm used by IdentifyFrobeniusGroup involves producing a hash value for the group based on evaluating a translate of the order class polynomial of the group at a particular point modulo a suitable prime number. This turns out to be a perfect hash for the Frobenius groups database, so no additional isomorphism tests are required to identify the group. (The specific prime and evaluation point are subject to change, and therefore are not documented.) For this reason, the use of the assign option can add considerably to the cost of identifying groups of larger order, as it requires an additional, explicit isomorphism computation.

By default the AllFrobeniusGroups( n ) command returns a list of all the Frobenius groups of order $n$, where $n$ is a positive integer for which the Frobenius groups of order $n$ are known. You can use the 'output' = "iterator" option to cause AllFrobeniusGroups to return an iterator object instead.

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

$\mathrm{NumFrobeniusGroups}\left(72\right)$

$2$

$\mathrm{G1}≔\mathrm{FrobeniusGroup}\left(72\,1\right)$

$\mathrm{G1}≔⟨\left(2\,3\,8\,9\,4\,6\,7\,5\right)\,\left(2\,8\,4\,7\right)\left(3\,9\,6\,5\right)\,\left(2\,4\right)\left(3\,6\right)\left(5\,9\right)\left(7\,8\right)\,\left(1\,2\,4\right)\left(3\,5\,7\right)\left(6\,8\,9\right)\,\left(1\,3\,6\right)\left(2\,5\,8\right)\left(4\,7\,9\right)⟩$

$\mathrm{IsFrobeniusGroup}\left(\mathrm{G1}\right)$

$\mathrm{true}$

$\mathrm{H1}≔\mathrm{FrobeniusComplement}\left(\mathrm{G1}\right)$

$\mathrm{H1}≔⟨\left(2\,3\,8\,9\,4\,6\,7\,5\right)\,\left(2\,8\,4\,7\right)\left(3\,9\,6\,5\right)\,\left(2\,4\right)\left(3\,6\right)\left(5\,9\right)\left(7\,8\right)⟩$

$\mathrm{IsMalnormal}\left(\mathrm{H1}\,\mathrm{G1}\right)$

$\mathrm{true}$

$\mathrm{G2}≔\mathrm{FrobeniusGroup}\left(\left[72\,2\right]\right)$

$\mathrm{G2}≔⟨\left(2\,8\,4\,7\right)\left(3\,9\,6\,5\right)\,\left(2\,3\,4\,6\right)\left(5\,7\,9\,8\right)\,\left(2\,4\right)\left(3\,6\right)\left(5\,9\right)\left(7\,8\right)\,\left(1\,2\,4\right)\left(3\,5\,7\right)\left(6\,8\,9\right)\,\left(1\,3\,6\right)\left(2\,5\,8\right)\left(4\,7\,9\right)⟩$

$\mathrm{AreIsomorphic}\left(\mathrm{G1}\,\mathrm{G2}\right)$

$\mathrm{false}$

$\mathrm{IdentifyFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(21\right)\right)$

$42,2$

$\mathrm{AreIsomorphic}\left(\mathrm{DihedralGroup}\left(21\right)\,\mathrm{FrobeniusGroup}\left(42\,2\right)\right)$

$\mathrm{true}$

$\mathrm{IdentifyFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(3^{10}\right)\right)$

Error, (in GroupTheory:-NumFrobeniusGroups) data for Frobenius groups of order 118098 not available

$\mathrm{IdentifyFrobeniusGroup}\left(\mathrm{CyclicGroup}\left(20\right)\right)$

Error, (in GroupTheory:-IdentifyFrobeniusGroup) group is not a Frobenius group

$\mathrm{IdentifyFrobeniusGroup}\left(\mathrm{Suzuki2B2}\left(2\right)\right)$

$20,1$

$L≔\mathrm{AllFrobeniusGroups}\left(100\right)$

$L≔\left[⟨\left(2\,8\,25\,21\right)\left(3\,6\,15\,10\right)\left(4\,20\,24\,9\right)\left(5\,18\,23\,12\right)\left(7\,19\,22\,11\right)\left(13\,16\,17\,14\right)\,\left(2\,25\right)\left(3\,15\right)\left(4\,24\right)\left(5\,23\right)\left(6\,10\right)\left(7\,22\right)\left(8\,21\right)\left(9\,20\right)\left(11\,19\right)\left(12\,18\right)\left(13\,17\right)\left(14\,16\right)\,\left(1\,2\,4\,7\,11\,3\,5\,8\,12\,16\,6\,9\,13\,17\,20\,10\,14\,18\,21\,23\,15\,19\,22\,24\,25\right)\,\left(1\,3\,6\,10\,15\right)\left(2\,5\,9\,14\,19\right)\left(4\,8\,13\,18\,22\right)\left(7\,12\,17\,21\,24\right)\left(11\,16\,20\,23\,25\right)⟩\,⟨\left(2\,4\,11\,7\right)\left(3\,6\,15\,10\right)\left(5\,13\,25\,21\right)\left(8\,20\,24\,14\right)\left(9\,22\,23\,12\right)\left(16\,17\,19\,18\right)\,\left(2\,11\right)\left(3\,15\right)\left(4\,7\right)\left(5\,25\right)\left(6\,10\right)\left(8\,24\right)\left(9\,23\right)\left(12\,22\right)\left(13\,21\right)\left(14\,20\right)\left(16\,19\right)\left(17\,18\right)\,\left(1\,2\,4\,7\,11\right)\left(3\,5\,8\,12\,16\right)\left(6\,9\,13\,17\,20\right)\left(10\,14\,18\,21\,23\right)\left(15\,19\,22\,24\,25\right)\,\left(1\,3\,6\,10\,15\right)\left(2\,5\,9\,14\,19\right)\left(4\,8\,13\,18\,22\right)\left(7\,12\,17\,21\,24\right)\left(11\,16\,20\,23\,25\right)⟩\,⟨\left(2\,7\,11\,4\right)\left(3\,6\,15\,10\right)\left(5\,17\,25\,18\right)\left(8\,9\,24\,23\right)\left(12\,20\,22\,14\right)\left(13\,19\,21\,16\right)\,\left(2\,11\right)\left(3\,15\right)\left(4\,7\right)\left(5\,25\right)\left(6\,10\right)\left(8\,24\right)\left(9\,23\right)\left(12\,22\right)\left(13\,21\right)\left(14\,20\right)\left(16\,19\right)\left(17\,18\right)\,\left(1\,2\,4\,7\,11\right)\left(3\,5\,8\,12\,16\right)\left(6\,9\,13\,17\,20\right)\left(10\,14\,18\,21\,23\right)\left(15\,19\,22\,24\,25\right)\,\left(1\,3\,6\,10\,15\right)\left(2\,5\,9\,14\,19\right)\left(4\,8\,13\,18\,22\right)\left(7\,12\,17\,21\,24\right)\left(11\,16\,20\,23\,25\right)⟩\right]$

$\mathrm{map}\left(\mathrm{IsFrobeniusGroup}\,L\right)$

$\left[\mathrm{true}\,\mathrm{true}\,\mathrm{true}\right]$

$\mathrm{cycomp}≔\mathrm{Array}\left(\left[\right]\right)\:$

$\mathrm{it}≔\mathrm{AllFrobeniusGroups}\left(15000\,'\mathrm{output}'=''iterator''\right)$

$\mathrm{it}≔\mathrm{\⟨Frobenius\; Groups\; Iterator:\; 15000/1\; ..\; 15000/12\⟩}$

$$\begin{gathered} \mathbf{for}\mathrm{id},G\mathbf{in}\mathrm{it}\mathbf{do} \\ \mathbf{if}\mathrm{IsCyclic}\left(\mathrm{FrobeniusComplement}\left(G\right)\right)\mathbf{then} \\ \mathrm{`,=`}\left(\mathrm{cycomp}\,\left[\mathrm{id}\right]\right) \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\mathrm{seq}\left(\mathrm{cycomp}\right)$

$\left[15000\,2\right],\left[15000\,6\right],\left[15000\,7\right],\left[15000\,8\right],\left[15000\,9\right]$

$G≔\mathrm{DihedralGroup}\left(333\right)\:$

$\mathrm{id}≔\mathrm{IdentifyFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(333\right)\,'\mathrm{assign}'='\mathrm{\eta }'\right)$

$\mathrm{id}≔666,2$

$\mathrm{F1}≔\mathrm{FrobeniusGroup}\left(\mathrm{id}\right)\:$

$\mathrm{F2}≔\mathrm{Image}\left(\mathrm{\eta }\right)$

$\mathrm{F2}≔⟨\left(1\,74\,121\,198\,253\,328\,44\,129\,176\,223\,308\,22\,99\,154\,231\,278\,30\,77\,124\,209\,256\,329\,55\,132\,179\,264\,311\,25\,110\,157\,234\,289\,33\,80\,165\,212\,259\,12\,58\,135\,190\,267\,314\,66\,113\,160\,245\,292\,36\,91\,168\,215\,300\,14\,61\,146\,193\,270\,324\,69\,116\,201\,248\,295\,47\,94\,171\,226\,303\,17\,102\,149\,196\,281\,2\,72\,127\,204\,251\,331\,50\,97\,182\,229\,306\,28\,105\,152\,237\,284\,3\,83\,130\,207\,262\,332\,53\,138\,185\,232\,317\,31\,108\,163\,240\,287\,39\,86\,133\,218\,265\,10\,64\,141\,188\,273\,320\,34\,119\,166\,243\,298\,42\,89\,174\,221\,268\,20\,67\,144\,199\,276\,322\,75\,122\,169\,254\,301\,45\,100\,177\,224\,309\,23\,70\,155\,202\,279\,5\,78\,125\,210\,257\,304\,56\,103\,180\,235\,312\,26\,111\,158\,205\,290\,6\,81\,136\,213\,260\,333\,59\,106\,191\,238\,315\,37\,114\,161\,246\,293\,8\,92\,139\,216\,271\,15\,62\,147\,194\,241\,325\,40\,117\,172\,249\,296\,48\,95\,142\,227\,274\,18\,73\,150\,197\,282\,327\,43\,128\,175\,252\,307\,51\,98\,183\,230\,277\,29\,76\,153\,208\,285\,4\,84\,131\,178\,263\,310\,54\,109\,186\,233\,318\,32\,79\,164\,211\,288\,11\,87\,134\,219\,266\,313\,65\,112\,189\,244\,321\,35\,120\,167\,214\,299\,13\,90\,145\,222\,269\,21\,68\,115\,200\,247\,323\,46\,123\,170\,255\,302\,16\,101\,148\,225\,280\,24\,71\,156\,203\,250\,330\,49\,126\,181\,258\,305\,57\,104\,151\,236\,283\,27\,82\,159\,206\,291\,7\,52\,137\,184\,261\,316\,60\,107\,192\,239\,286\,38\,85\,162\,217\,294\,9\,93\,140\,187\,272\,319\,63\,118\,195\,242\,326\,41\,88\,173\,220\,297\,19\,96\,143\,228\,275\right)\,\left(1\,4\right)\left(2\,5\right)\left(3\,322\right)\left(6\,324\right)\left(7\,21\right)\left(8\,314\right)\left(9\,313\right)\left(11\,319\right)\left(12\,15\right)\left(13\,316\right)\left(14\,333\right)\left(16\,305\right)\left(17\,304\right)\left(18\,329\right)\left(19\,310\right)\left(20\,332\right)\left(22\,307\right)\left(23\,331\right)\left(24\,330\right)\left(25\,296\right)\left(26\,295\right)\left(27\,323\right)\left(28\,301\right)\left(29\,328\right)\left(30\,327\right)\left(31\,298\right)\left(32\,326\right)\left(33\,325\right)\left(34\,287\right)\left(35\,286\right)\left(36\,315\right)\left(37\,292\right)\left(38\,321\right)\left(39\,320\right)\left(40\,289\right)\left(41\,318\right)\left(42\,317\right)\left(43\,278\right)\left(44\,277\right)\left(45\,306\right)\left(46\,283\right)\left(47\,312\right)\left(48\,311\right)\left(49\,280\right)\left(50\,309\right)\left(51\,308\right)\left(52\,269\right)\left(53\,268\right)\left(54\,297\right)\left(55\,274\right)\left(56\,303\right)\left(57\,302\right)\left(58\,271\right)\left(59\,300\right)\left(60\,299\right)\left(61\,260\right)\left(62\,259\right)\left(63\,288\right)\left(64\,265\right)\left(65\,294\right)\left(66\,293\right)\left(67\,262\right)\left(68\,291\right)\left(69\,290\right)\left(70\,251\right)\left(71\,250\right)\left(72\,279\right)\left(73\,256\right)\left(74\,285\right)\left(75\,284\right)\left(76\,253\right)\left(77\,282\right)\left(78\,281\right)\left(79\,242\right)\left(80\,241\right)\left(81\,270\right)\left(82\,247\right)\left(83\,276\right)\left(84\,275\right)\left(85\,244\right)\left(86\,273\right)\left(87\,272\right)\left(88\,233\right)\left(89\,232\right)\left(90\,261\right)\left(91\,238\right)\left(92\,267\right)\left(93\,266\right)\left(94\,235\right)\left(95\,264\right)\left(96\,263\right)\left(97\,224\right)\left(98\,223\right)\left(99\,252\right)\left(100\,229\right)\left(101\,258\right)\left(102\,257\right)\left(103\,226\right)\left(104\,255\right)\left(105\,254\right)\left(106\,215\right)\left(107\,214\right)\left(108\,243\right)\left(109\,220\right)\left(110\,249\right)\left(111\,248\right)\left(112\,217\right)\left(113\,246\right)\left(114\,245\right)\left(115\,206\right)\left(116\,205\right)\left(117\,234\right)\left(118\,211\right)\left(119\,240\right)\left(120\,239\right)\left(121\,208\right)\left(122\,237\right)\left(123\,236\right)\left(124\,197\right)\left(125\,196\right)\left(126\,225\right)\left(127\,202\right)\left(128\,231\right)\left(129\,230\right)\left(130\,199\right)\left(131\,228\right)\left(132\,227\right)\left(133\,188\right)\left(134\,187\right)\left(135\,216\right)\left(136\,193\right)\left(137\,222\right)\left(138\,221\right)\left(139\,190\right)\left(140\,219\right)\left(141\,218\right)\left(142\,179\right)\left(143\,178\right)\left(144\,207\right)\left(145\,184\right)\left(146\,213\right)\left(147\,212\right)\left(148\,181\right)\left(149\,210\right)\left(150\,209\right)\left(151\,170\right)\left(152\,169\right)\left(153\,198\right)\left(154\,175\right)\left(155\,204\right)\left(156\,203\right)\left(157\,172\right)\left(158\,201\right)\left(159\,200\right)\left(160\,161\right)\left(162\,189\right)\left(163\,166\right)\left(164\,195\right)\left(165\,194\right)\left(167\,192\right)\left(168\,191\right)\left(171\,180\right)\left(173\,186\right)\left(174\,185\right)\left(176\,183\right)\left(177\,182\right)⟩$

$\mathrm{IsSubgroup}\left(\mathrm{F1}\,\mathrm{F2}\right)\mathbf{and}\mathrm{IsSubgroup}\left(\mathrm{F2}\,\mathrm{F1}\right)$

$\mathrm{true}$

$\mathrm{id}≔\mathrm{SearchFrobeniusGroups}\left('\mathrm{soluble}'=\mathrm{false}\right)$

$\mathrm{id}≔\left[14520\,2\right]$

$G≔\mathrm{FrobeniusGroup}\left(\mathrm{id}\right)$

$G≔⟨\left(2\,6\,74\,80\,16\right)\left(3\,49\,63\,116\,19\right)\left(4\,51\,108\,68\,59\right)\left(5\,53\,52\,110\,40\right)\left(7\,10\,37\,47\,78\right)\left(8\,17\,61\,60\,90\right)\left(9\,20\,89\,97\,93\right)\left(11\,43\,106\,48\,31\right)\left(12\,103\,70\,29\,36\right)\left(13\,104\,85\,86\,84\right)\left(14\,81\,38\,42\,41\right)\left(15\,58\,64\,79\,77\right)\left(18\,105\,56\,101\,91\right)\left(21\,102\,109\,54\,112\right)\left(22\,26\,100\,55\,32\right)\left(23\,39\,28\,33\,35\right)\left(24\,83\,98\,71\,75\right)\left(25\,118\,76\,119\,95\right)\left(27\,30\,117\,94\,92\right)\left(34\,99\,120\,69\,72\right)\left(44\,62\,113\,87\,82\right)\left(45\,121\,88\,66\,57\right)\left(46\,115\,107\,114\,65\right)\left(50\,111\,67\,96\,73\right)\,\left(2\,22\,21\,99\,104\right)\left(3\,58\,75\,73\,20\right)\left(4\,26\,19\,65\,78\right)\left(5\,64\,16\,92\,36\right)\left(6\,9\,91\,37\,17\right)\left(7\,94\,85\,96\,106\right)\left(8\,52\,54\,57\,63\right)\left(10\,23\,102\,15\,38\right)\left(11\,95\,93\,87\,109\right)\left(12\,114\,89\,34\,39\right)\left(13\,119\,70\,81\,49\right)\left(14\,108\,98\,113\,74\right)\left(18\,84\,45\,83\,35\right)\left(24\,32\,61\,103\,43\right)\left(25\,33\,68\,90\,30\right)\left(27\,116\,62\,69\,56\right)\left(28\,67\,44\,100\,40\right)\left(29\,101\,50\,112\,51\right)\left(31\,110\,41\,115\,105\right)\left(42\,55\,121\,97\,117\right)\left(46\,80\,66\,111\,118\right)\left(47\,76\,120\,71\,53\right)\left(48\,72\,88\,79\,59\right)\left(60\,77\,82\,86\,107\right)\,\left(2\,3\right)\left(4\,5\right)\left(6\,49\right)\left(7\,12\right)\left(8\,14\right)\left(9\,13\right)\left(10\,103\right)\left(11\,35\right)\left(15\,32\right)\left(16\,19\right)\left(17\,81\right)\left(18\,95\right)\left(20\,104\right)\left(21\,75\right)\left(22\,58\right)\left(23\,43\right)\left(24\,102\right)\left(25\,105\right)\left(26\,64\right)\left(27\,46\right)\left(28\,48\right)\left(29\,47\right)\left(30\,115\right)\left(31\,33\right)\left(34\,96\right)\left(36\,78\right)\left(37\,70\right)\left(38\,61\right)\left(39\,106\right)\left(40\,59\right)\left(41\,90\right)\left(42\,60\right)\left(44\,88\right)\left(45\,87\right)\left(50\,120\right)\left(51\,53\right)\left(52\,108\right)\left(54\,98\right)\left(55\,77\right)\left(56\,118\right)\left(57\,113\right)\left(62\,66\right)\left(63\,74\right)\left(65\,92\right)\left(67\,72\right)\left(68\,110\right)\left(69\,111\right)\left(71\,112\right)\left(73\,99\right)\left(76\,101\right)\left(79\,100\right)\left(80\,116\right)\left(82\,121\right)\left(83\,109\right)\left(84\,93\right)\left(85\,89\right)\left(86\,97\right)\left(91\,119\right)\left(94\,114\right)\left(107\,117\right)\,\left(1\,2\,7\,27\,79\,53\,51\,100\,46\,12\,3\right)\left(4\,8\,28\,72\,52\,84\,70\,112\,102\,47\,13\right)\left(5\,9\,29\,24\,71\,37\,93\,108\,67\,48\,14\right)\left(6\,22\,69\,97\,66\,20\,17\,15\,19\,35\,23\right)\left(10\,38\,94\,107\,50\,36\,33\,87\,76\,82\,39\right)\left(11\,16\,32\,81\,104\,62\,86\,111\,58\,49\,43\right)\left(18\,59\,75\,110\,105\,99\,44\,26\,77\,113\,60\right)\left(21\,40\,95\,42\,57\,55\,64\,88\,73\,25\,68\right)\left(30\,85\,65\,116\,118\,83\,34\,90\,98\,119\,74\right)\left(31\,78\,120\,117\,114\,61\,103\,106\,121\,101\,45\right)\left(41\,96\,109\,56\,80\,92\,89\,115\,63\,91\,54\right)\,\left(1\,4\,16\,26\,78\,65\,92\,36\,64\,19\,5\right)\left(2\,8\,32\,77\,120\,116\,89\,33\,88\,35\,9\right)\left(3\,13\,11\,44\,31\,85\,80\,50\,55\,15\,14\right)\left(6\,24\,27\,72\,104\,60\,114\,83\,63\,76\,25\right)\left(7\,28\,81\,113\,117\,118\,115\,87\,73\,23\,29\right)\left(10\,40\,97\,93\,51\,70\,111\,75\,106\,98\,41\right)\left(12\,47\,43\,99\,45\,30\,56\,107\,57\,17\,48\right)\left(18\,61\,34\,91\,82\,68\,22\,71\,79\,52\,62\right)\left(20\,67\,46\,102\,49\,105\,101\,74\,109\,94\,42\right)\left(21\,69\,37\,53\,84\,86\,59\,103\,90\,54\,39\right)\left(38\,95\,66\,108\,100\,112\,58\,110\,121\,119\,96\right)⟩$

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

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{FrobeniusComplement}\left(G\right)\,\mathrm{SL}\left(2\,5\right)\right)$

$\mathrm{true}$

The GroupTheory[FrobeniusGroup], GroupTheory[NumFrobeniusGroups], GroupTheory[IdentifyFrobeniusGroup] and GroupTheory[AllFrobeniusGroups] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.