Dic₉₉₂₄ #72274044

order	39696
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₉₉₂₄

Splittings

Hall	C₃×C₈₂₇ ⋊ Q₁₆
Sylow	C₈₂₇ ⋊ Dic₁₂
Sylow	C₃ ⋊ Dic₃₃₀₈

Subgroups

	subgroup	G/H
center	C₂	D₉₉₂₄
commutator, cosocle	C₃×C₄×C₈₂₇	C₂²
Frattini	C₄	C₂×D₂₄₈₁
Fitting	C₃×C₈×C₈₂₇	C₂
socle, prime radical	C₂×C₃×C₈₂₇	D₄
hypercenter	C₈	D₂₄₈₁
hypocenter	C₃×C₈₂₇	Q₁₆
LCS	G ▷ C₃×C₄×C₈₂₇ ▷ C₂×C₃×C₈₂₇ ▷ C₃×C₈₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 846
transitive degree	μ_t(G) = 39696

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

Polycyclic presentation:

2^4,3,827|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^826

Conjugacy classes & character table

9927 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, 827]);;
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.6^826);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic9924 #72274044