D₁₁₅₃₉ #37528099

order	23078
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₅₃₉
unambiguous semidirect product	C₁₁⋊D₁₀₄₉
unambiguous semidirect product	C₁₀₄₉⋊D₁₁

Splittings

Fitting, socle, derived, Hall	C₁₁×C₁₀₄₉ ⋊ C₂
Sylow	C₁₀₄₉ ⋊ D₁₁
Sylow	C₁₁ ⋊ D₁₀₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₁₀₄₉	C₂
LCS	G ▷ C₁₁×C₁₀₄₉
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1060
transitive degree	μ_t(G) = 11539

A minimal faithful permutation representation — 2 generators on 1060 points:

(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060);(2,11)(3,10)(4,9)(5,8)(6,7)(13,1060)(14,1059)(15,1058)(16,1057)(17,1056)(18,1055)(19,1054)(20,1053)(21,1052)(22,1051)(23,1050)(24,1049)(25,1048)(26,1047)(27,1046)(28,1045)(29,1044)(30,1043)(31,1042)(32,1041)(33,1040)(34,1039)(35,1038)(36,1037)(37,1036)(38,1035)(39,1034)(40,1033)(41,1032)(42,1031)(43,1030)(44,1029)(45,1028)(46,1027)(47,1026)(48,1025)(49,1024)(50,1023)(51,1022)(52,1021)(53,1020)(54,1019)(55,1018)(56,1017)(57,1016)(58,1015)(59,1014)(60,1013)(61,1012)(62,1011)(63,1010)(64,1009)(65,1008)(66,1007)(67,1006)(68,1005)(69,1004)(70,1003)(71,1002)(72,1001)(73,1000)(74,999)(75,998)(76,997)(77,996)(78,995)(79,994)(80,993)(81,992)(82,991)(83,990)(84,989)(85,988)(86,987)(87,986)(88,985)(89,984)(90,983)(91,982)(92,981)(93,980)(94,979)(95,978)(96,977)(97,976)(98,975)(99,974)(100,973)(101,972)(102,971)(103,970)(104,969)(105,968)(106,967)(107,966)(108,965)(109,964)(110,963)(111,962)(112,961)(113,960)(114,959)(115,958)(116,957)(117,956)(118,955)(119,954)(120,953)(121,952)(122,951)(123,950)(124,949)(125,948)(126,947)(127,946)(128,945)(129,944)(130,943)(131,942)(132,941)(133,940)(134,939)(135,938)(136,937)(137,936)(138,935)(139,934)(140,933)(141,932)(142,931)(143,930)(144,929)(145,928)(146,927)(147,926)(148,925)(149,924)(150,923)(151,922)(152,921)(153,920)(154,919)(155,918)(156,917)(157,916)(158,915)(159,914)(160,913)(161,912)(162,911)(163,910)(164,909)(165,908)(166,907)(167,906)(168,905)(169,904)(170,903)(171,902)(172,901)(173,900)(174,899)(175,898)(176,897)(177,896)(178,895)(179,894)(180,893)(181,892)(182,891)(183,890)(184,889)(185,888)(186,887)(187,886)(188,885)(189,884)(190,883)(191,882)(192,881)(193,880)(194,879)(195,878)(196,877)(197,876)(198,875)(199,874)(200,873)(201,872)(202,871)(203,870)(204,869)(205,868)(206,867)(207,866)(208,865)(209,864)(210,863)(211,862)(212,861)(213,860)(214,859)(215,858)(216,857)(217,856)(218,855)(219,854)(220,853)(221,852)(222,851)(223,850)(224,849)(225,848)(226,847)(227,846)(228,845)(229,844)(230,843)(231,842)(232,841)(233,840)(234,839)(235,838)(236,837)(237,836)(238,835)(239,834)(240,833)(241,832)(242,831)(243,830)(244,829)(245,828)(246,827)(247,826)(248,825)(249,824)(250,823)(251,822)(252,821)(253,820)(254,819)(255,818)(256,817)(257,816)(258,815)(259,814)(260,813)(261,812)(262,811)(263,810)(264,809)(265,808)(266,807)(267,806)(268,805)(269,804)(270,803)(271,802)(272,801)(273,800)(274,799)(275,798)(276,797)(277,796)(278,795)(279,794)(280,793)(281,792)(282,791)(283,790)(284,789)(285,788)(286,787)(287,786)(288,785)(289,784)(290,783)(291,782)(292,781)(293,780)(294,779)(295,778)(296,777)(297,776)(298,775)(299,774)(300,773)(301,772)(302,771)(303,770)(304,769)(305,768)(306,767)(307,766)(308,765)(309,764)(310,763)(311,762)(312,761)(313,760)(314,759)(315,758)(316,757)(317,756)(318,755)(319,754)(320,753)(321,752)(322,751)(323,750)(324,749)(325,748)(326,747)(327,746)(328,745)(329,744)(330,743)(331,742)(332,741)(333,740)(334,739)(335,738)(336,737)(337,736)(338,735)(339,734)(340,733)(341,732)(342,731)(343,730)(344,729)(345,728)(346,727)(347,726)(348,725)(349,724)(350,723)(351,722)(352,721)(353,720)(354,719)(355,718)(356,717)(357,716)(358,715)(359,714)(360,713)(361,712)(362,711)(363,710)(364,709)(365,708)(366,707)(367,706)(368,705)(369,704)(370,703)(371,702)(372,701)(373,700)(374,699)(375,698)(376,697)(377,696)(378,695)(379,694)(380,693)(381,692)(382,691)(383,690)(384,689)(385,688)(386,687)(387,686)(388,685)(389,684)(390,683)(391,682)(392,681)(393,680)(394,679)(395,678)(396,677)(397,676)(398,675)(399,674)(400,673)(401,672)(402,671)(403,670)(404,669)(405,668)(406,667)(407,666)(408,665)(409,664)(410,663)(411,662)(412,661)(413,660)(414,659)(415,658)(416,657)(417,656)(418,655)(419,654)(420,653)(421,652)(422,651)(423,650)(424,649)(425,648)(426,647)(427,646)(428,645)(429,644)(430,643)(431,642)(432,641)(433,640)(434,639)(435,638)(436,637)(437,636)(438,635)(439,634)(440,633)(441,632)(442,631)(443,630)(444,629)(445,628)(446,627)(447,626)(448,625)(449,624)(450,623)(451,622)(452,621)(453,620)(454,619)(455,618)(456,617)(457,616)(458,615)(459,614)(460,613)(461,612)(462,611)(463,610)(464,609)(465,608)(466,607)(467,606)(468,605)(469,604)(470,603)(471,602)(472,601)(473,600)(474,599)(475,598)(476,597)(477,596)(478,595)(479,594)(480,593)(481,592)(482,591)(483,590)(484,589)(485,588)(486,587)(487,586)(488,585)(489,584)(490,583)(491,582)(492,581)(493,580)(494,579)(495,578)(496,577)(497,576)(498,575)(499,574)(500,573)(501,572)(502,571)(503,570)(504,569)(505,568)(506,567)(507,566)(508,565)(509,564)(510,563)(511,562)(512,561)(513,560)(514,559)(515,558)(516,557)(517,556)(518,555)(519,554)(520,553)(521,552)(522,551)(523,550)(524,549)(525,548)(526,547)(527,546)(528,545)(529,544)(530,543)(531,542)(532,541)(533,540)(534,539)(535,538)(536,537)

Polycyclic presentation:

2,11,1049|b^a=b^10,c^a=c^1048

Conjugacy classes & character table

5771 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23078, 3);

D11539 #37528099