D₂₃₇₈₄ #72283204

order	47568
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₃₇₈₄

Splittings

Fitting	C₃×C₈×C₉₉₁ ⋊ C₂
Hall	C₃×C₉₉₁ ⋊ D₈
Sylow	C₉₉₁ ⋊ D₂₄
Sylow	C₃ ⋊ D₇₉₂₈

Subgroups

	subgroup	G/H
center	C₂	D₁₁₈₉₂
commutator, cosocle	C₃×C₄×C₉₉₁	C₂²
Frattini	C₄	C₂×D₂₉₇₃
Fitting	C₃×C₈×C₉₉₁	C₂
socle	C₂×C₃×C₉₉₁	D₄
hypercenter	C₈	D₂₉₇₃
hypocenter	C₃×C₉₉₁	D₈
LCS	G ▷ C₃×C₄×C₉₉₁ ▷ C₂×C₃×C₉₉₁ ▷ C₃×C₉₉₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1002
transitive degree	μ_t(G) = 23784

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

Polycyclic presentation:

2^4,3,991|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^990

Conjugacy classes & character table

11895 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 991]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^990);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D23784 #72283204