Dic₁₀₉₃₂ #72275019

order	43728
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₉₃₂

Splittings

Hall	C₃×C₉₁₁ ⋊ Q₁₆
Sylow	C₉₁₁ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₃₆₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₀₉₃₂
commutator, cosocle	C₃×C₄×C₉₁₁	C₂²
Frattini	C₄	C₂×D₂₇₃₃
Fitting	C₃×C₈×C₉₁₁	C₂
socle, prime radical	C₂×C₃×C₉₁₁	D₄
hypercenter	C₈	D₂₇₃₃
hypocenter	C₃×C₉₁₁	Q₁₆
LCS	G ▷ C₃×C₄×C₉₁₁ ▷ C₂×C₃×C₉₁₁ ▷ C₃×C₉₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 930
transitive degree	μ_t(G) = 43728

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

Polycyclic presentation:

2^4,3,911|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^910

Conjugacy classes & character table

10935 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, 911]);;
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.6^910);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic10932 #72275019