D₁₁₂₃₁ #36376384

order	22462
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₂₃₁
unambiguous semidirect product	C₁₁⋊D₁₀₂₁
unambiguous semidirect product	C₁₀₂₁⋊D₁₁

Splittings

Fitting, socle, derived, Hall	C₁₁×C₁₀₂₁ ⋊ C₂
Sylow	C₁₀₂₁ ⋊ D₁₁
Sylow	C₁₁ ⋊ D₁₀₂₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₁₀₂₁	C₂
LCS	G ▷ C₁₁×C₁₀₂₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1032
transitive degree	μ_t(G) = 11231

A minimal faithful permutation representation — 2 generators on 1032 points:

(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032);(2,11)(3,10)(4,9)(5,8)(6,7)(13,1032)(14,1031)(15,1030)(16,1029)(17,1028)(18,1027)(19,1026)(20,1025)(21,1024)(22,1023)(23,1022)(24,1021)(25,1020)(26,1019)(27,1018)(28,1017)(29,1016)(30,1015)(31,1014)(32,1013)(33,1012)(34,1011)(35,1010)(36,1009)(37,1008)(38,1007)(39,1006)(40,1005)(41,1004)(42,1003)(43,1002)(44,1001)(45,1000)(46,999)(47,998)(48,997)(49,996)(50,995)(51,994)(52,993)(53,992)(54,991)(55,990)(56,989)(57,988)(58,987)(59,986)(60,985)(61,984)(62,983)(63,982)(64,981)(65,980)(66,979)(67,978)(68,977)(69,976)(70,975)(71,974)(72,973)(73,972)(74,971)(75,970)(76,969)(77,968)(78,967)(79,966)(80,965)(81,964)(82,963)(83,962)(84,961)(85,960)(86,959)(87,958)(88,957)(89,956)(90,955)(91,954)(92,953)(93,952)(94,951)(95,950)(96,949)(97,948)(98,947)(99,946)(100,945)(101,944)(102,943)(103,942)(104,941)(105,940)(106,939)(107,938)(108,937)(109,936)(110,935)(111,934)(112,933)(113,932)(114,931)(115,930)(116,929)(117,928)(118,927)(119,926)(120,925)(121,924)(122,923)(123,922)(124,921)(125,920)(126,919)(127,918)(128,917)(129,916)(130,915)(131,914)(132,913)(133,912)(134,911)(135,910)(136,909)(137,908)(138,907)(139,906)(140,905)(141,904)(142,903)(143,902)(144,901)(145,900)(146,899)(147,898)(148,897)(149,896)(150,895)(151,894)(152,893)(153,892)(154,891)(155,890)(156,889)(157,888)(158,887)(159,886)(160,885)(161,884)(162,883)(163,882)(164,881)(165,880)(166,879)(167,878)(168,877)(169,876)(170,875)(171,874)(172,873)(173,872)(174,871)(175,870)(176,869)(177,868)(178,867)(179,866)(180,865)(181,864)(182,863)(183,862)(184,861)(185,860)(186,859)(187,858)(188,857)(189,856)(190,855)(191,854)(192,853)(193,852)(194,851)(195,850)(196,849)(197,848)(198,847)(199,846)(200,845)(201,844)(202,843)(203,842)(204,841)(205,840)(206,839)(207,838)(208,837)(209,836)(210,835)(211,834)(212,833)(213,832)(214,831)(215,830)(216,829)(217,828)(218,827)(219,826)(220,825)(221,824)(222,823)(223,822)(224,821)(225,820)(226,819)(227,818)(228,817)(229,816)(230,815)(231,814)(232,813)(233,812)(234,811)(235,810)(236,809)(237,808)(238,807)(239,806)(240,805)(241,804)(242,803)(243,802)(244,801)(245,800)(246,799)(247,798)(248,797)(249,796)(250,795)(251,794)(252,793)(253,792)(254,791)(255,790)(256,789)(257,788)(258,787)(259,786)(260,785)(261,784)(262,783)(263,782)(264,781)(265,780)(266,779)(267,778)(268,777)(269,776)(270,775)(271,774)(272,773)(273,772)(274,771)(275,770)(276,769)(277,768)(278,767)(279,766)(280,765)(281,764)(282,763)(283,762)(284,761)(285,760)(286,759)(287,758)(288,757)(289,756)(290,755)(291,754)(292,753)(293,752)(294,751)(295,750)(296,749)(297,748)(298,747)(299,746)(300,745)(301,744)(302,743)(303,742)(304,741)(305,740)(306,739)(307,738)(308,737)(309,736)(310,735)(311,734)(312,733)(313,732)(314,731)(315,730)(316,729)(317,728)(318,727)(319,726)(320,725)(321,724)(322,723)(323,722)(324,721)(325,720)(326,719)(327,718)(328,717)(329,716)(330,715)(331,714)(332,713)(333,712)(334,711)(335,710)(336,709)(337,708)(338,707)(339,706)(340,705)(341,704)(342,703)(343,702)(344,701)(345,700)(346,699)(347,698)(348,697)(349,696)(350,695)(351,694)(352,693)(353,692)(354,691)(355,690)(356,689)(357,688)(358,687)(359,686)(360,685)(361,684)(362,683)(363,682)(364,681)(365,680)(366,679)(367,678)(368,677)(369,676)(370,675)(371,674)(372,673)(373,672)(374,671)(375,670)(376,669)(377,668)(378,667)(379,666)(380,665)(381,664)(382,663)(383,662)(384,661)(385,660)(386,659)(387,658)(388,657)(389,656)(390,655)(391,654)(392,653)(393,652)(394,651)(395,650)(396,649)(397,648)(398,647)(399,646)(400,645)(401,644)(402,643)(403,642)(404,641)(405,640)(406,639)(407,638)(408,637)(409,636)(410,635)(411,634)(412,633)(413,632)(414,631)(415,630)(416,629)(417,628)(418,627)(419,626)(420,625)(421,624)(422,623)(423,622)(424,621)(425,620)(426,619)(427,618)(428,617)(429,616)(430,615)(431,614)(432,613)(433,612)(434,611)(435,610)(436,609)(437,608)(438,607)(439,606)(440,605)(441,604)(442,603)(443,602)(444,601)(445,600)(446,599)(447,598)(448,597)(449,596)(450,595)(451,594)(452,593)(453,592)(454,591)(455,590)(456,589)(457,588)(458,587)(459,586)(460,585)(461,584)(462,583)(463,582)(464,581)(465,580)(466,579)(467,578)(468,577)(469,576)(470,575)(471,574)(472,573)(473,572)(474,571)(475,570)(476,569)(477,568)(478,567)(479,566)(480,565)(481,564)(482,563)(483,562)(484,561)(485,560)(486,559)(487,558)(488,557)(489,556)(490,555)(491,554)(492,553)(493,552)(494,551)(495,550)(496,549)(497,548)(498,547)(499,546)(500,545)(501,544)(502,543)(503,542)(504,541)(505,540)(506,539)(507,538)(508,537)(509,536)(510,535)(511,534)(512,533)(513,532)(514,531)(515,530)(516,529)(517,528)(518,527)(519,526)(520,525)(521,524)(522,523)

Polycyclic presentation:

2,11,1021|b^a=b^10,c^a=c^1020

Conjugacy classes & character table

5617 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22462, 3);

D11231 #36376384