Let $G$ and $H$ be permutation groups. The wreath product G $\wr$ H_ of $G$ by $H$ is a permutation group constructed as a semi-direct product of $d$ copies of $G$ (called the base group), where $d$ is the degree of $H$, and the action of $H$ on the base group is the action of $H$ by permuting the copies of $G$. Thus, the order of G $\wr$ H_ is equal to ${\left|G\right|}^d\left|H\right|$, and the degree of the wreath product is the product of the degrees of $G$ and $H$.

The regular wreath product of $G$ and $H$ is the wreath product in which $H$ is considered as a regular permutation group on itself. (This is also called the "standard wreath product".)

The WreathProduct( G, H ) command returns a permutation group that is the wreath product G $\wr$ H_.

If more than two groups are provided as input, then an iterated wreath product is constructed using the left associative rule. For example, WreathProduct( A, B, C, D ) returns ((A $\wr$ B) $\wr$ C) $\wr$ D_.

The RegularWreathProduct( G, H ) command returns the regular wreath product of $G$ and $H$. In this case, it is not required that $H$ be a permutation group, as a regular permutation representation of the finite group $H$ is used instead. (Here, $H$ may be either a Cayley table group or a finitely presented finite group, as well as a permutation group, which need not be itself regular.)

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

$G≔\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(2\right)\,\mathrm{CyclicGroup}\left(2\right)\right)$

$G≔⟨\left(1\,2\right)\,\left(1\,3\right)\left(2\,4\right)⟩$

$\mathrm{AreIsomorphic}\left(G\,\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\right)\,\mathrm{SylowSubgroup}\left(3\,\mathrm{Symm}\left(9\right)\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{WreathProduct}\left(\mathrm{`\$`}\left(\mathrm{CyclicGroup}\left(2\right)\,3\right)\right)\,\mathrm{SylowSubgroup}\left(2\,\mathrm{Symm}\left(2^3\right)\right)\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\right)\right)$

$81$

$\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\right)\right)$

$1594323$

$\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\right)\right)$

$12157665459056928801$

$\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\,\mathrm{CyclicGroup}\left(3\right)\right)\right)$

$5391030899743293631239539488528815119194426882613553319203$

$W≔\mathrm{WreathProduct}\left(\mathrm{Alt}\left(4\right)\,\mathrm{Symm}\left(3\right)\right)$

$W≔⟨\left(1\,2\,3\right)\,\left(2\,3\,4\right)\,\left(1\,5\right)\left(2\,6\right)\left(3\,7\right)\left(4\,8\right)\,\left(1\,5\,9\right)\left(2\,6\,10\right)\left(3\,7\,11\right)\left(4\,8\,12\right)⟩$

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

$10368$

$\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{Symm}\left(3\right)\,\mathrm{Alt}\left(4\right)\right)\right)$

$15552$

$\mathrm{GroupOrder}\left(\mathrm{RegularWreathProduct}\left(\mathrm{Alt}\left(4\right)\,\mathrm{Symm}\left(3\right)\right)\right)$

$17915904$

$\mathrm{IsTransitive}\left(W\right)$

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{BlockSystem}\left(W\right)$

$\left\{\left\{1\,2\,3\,4\right\}\,\left\{5\,6\,7\,8\right\}\,\left\{9\,10\,11\,12\right\}\right\}$

$W≔\mathrm{WreathProduct}\left(\mathrm{Alt}\left(4\right)\,\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)\right]\right)\right)\:$

$\mathrm{IsTransitive}\left(W\right)$

$\mathrm{false}$

$\mathrm{map}\left(\mathrm{Elements}\,\mathrm{Orbits}\left(W\right)\right)$

$\left[\left\{1\,2\,3\,4\,5\,6\,7\,8\right\}\,\left\{9\,10\,11\,12\,13\,14\,15\,16\right\}\right]$

$W≔\mathrm{WreathProduct}\left(\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)\right]\right)\,\mathrm{Symm}\left(3\right)\right)\:$

$\mathrm{IsTransitive}\left(W\right)$

$\mathrm{false}$

$\mathrm{map}\left(\mathrm{Elements}\,\mathrm{Orbits}\left(W\right)\right)$

$\left[\left\{1\,2\,5\,6\,9\,10\right\}\,\left\{3\,4\,7\,8\,11\,12\right\}\right]$

$W≔\mathrm{RegularWreathProduct}\left(\mathrm{CyclicGroup}\left(2\right)\,⟨⟨a\,b⟩|⟨a^2\,b^3\,{\left(a·b\right)}^5=1⟩⟩\right)$

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

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

$\mathrm{false}$

$\mathrm{BlockSystem}\left(W\right)$

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

$S≔\mathrm{Stabilizer}\left(109\,W\right)\:$

$\mathrm{Index}\left(S\,W\right)$

$120$

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

$576460752303423488$

$\mathrm{IsElementary}\left(S\right)$

$\mathrm{true}$

$\mathrm{DirectFactors}\left(S\right)$

$⟨\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)\,\left(119\,120\right)⟩,⟨\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)\,\left(117\,118\right)⟩,⟨\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)\,\left(115\,116\right)⟩,⟨\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)\,\left(113\,114\right)⟩,⟨\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)\,\left(111\,112\right)⟩,⟨\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)\,\left(107\,108\right)⟩,⟨\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)\,\left(105\,106\right)⟩,⟨\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)\,\left(103\,104\right)⟩,⟨\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)\,\left(101\,102\right)⟩,⟨\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)\,\left(99\,100\right)⟩,⟨\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)\,\left(97\,98\right)⟩,⟨\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)\,\left(95\,96\right)⟩,⟨\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)\,\left(93\,94\right)⟩,⟨\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)\,\left(91\,92\right)⟩,⟨\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)\,\left(89\,90\right)⟩,⟨\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\left(87\,88\right)\,\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ght)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)\,\left(69\,70\right)⟩,⟨\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)\,\left(67\,68\right)⟩,⟨\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)\,\left(65\,66\right)⟩,⟨\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)\,\left(63\,64\right)⟩,⟨\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)\,\left(61\,62\right)⟩,⟨\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)\,\left(59\,60\right)⟩,⟨\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)\,\left(29\,30\right)⟩,⟨\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)\,\left(27\,28\right)⟩,⟨\left(25\,26\right)\,\left(25\,26\right)\,\left(25\,26\right)\,\left(25\,26\right)\,\left(25\,26\right)\,\left(25\,26\right)\,\left(25\,26\right)\,\left(25\,26\right)\,\left(25\,26\right)\,\left(25\,26\right)\,\left(25\,26\right)\,\left(25\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ight)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)\,\left(39\,40\right)⟩,⟨\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)\,\left(41\,42\right)⟩,⟨\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)\,\left(43\,44\right)⟩,⟨\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)\,\left(45\,46\right)⟩,⟨\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)\,\left(47\,48\right)⟩,⟨\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)\,\left(49\,50\right)⟩,⟨\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)\,\left(51\,52\right)⟩,⟨\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)\,\left(53\,54\right)⟩,⟨\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55\,56\right)\,\left(55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