Dic₁₁₂₄₄ #72275333

order	44976
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₂₄₄

Splittings

Hall	C₃×C₉₃₇ ⋊ Q₁₆
Sylow	C₉₃₇ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₃₇₄₈

Subgroups

	subgroup	G/H
center	C₂	D₁₁₂₄₄
commutator, cosocle	C₃×C₄×C₉₃₇	C₂²
Frattini	C₄	C₂×D₂₈₁₁
Fitting	C₃×C₈×C₉₃₇	C₂
socle, prime radical	C₂×C₃×C₉₃₇	D₄
hypercenter	C₈	D₂₈₁₁
hypocenter	C₃×C₉₃₇	Q₁₆
LCS	G ▷ C₃×C₄×C₉₃₇ ▷ C₂×C₃×C₉₃₇ ▷ C₃×C₉₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 956
transitive degree	μ_t(G) = 44976

A minimal faithful permutation representation — 2 generators on 956 points:

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

Polycyclic presentation:

2^4,3,937|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^936

Conjugacy classes & character table

11247 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, 937]);;
gap> SetPower(coll, 1, F.4);;
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.6^936);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic11244 #72275333