D₁₉₄₆₄ #72274016

order	38928
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₉₄₆₄

Splittings

Fitting	C₃×C₈×C₈₁₁ ⋊ C₂
Hall	C₃×C₈₁₁ ⋊ D₈
Sylow	C₈₁₁ ⋊ D₂₄
Sylow	C₃ ⋊ D₆₄₈₈

Subgroups

	subgroup	G/H
center	C₂	D₉₇₃₂
commutator, cosocle	C₃×C₄×C₈₁₁	C₂²
Frattini	C₄	C₂×D₂₄₃₃
Fitting	C₃×C₈×C₈₁₁	C₂
socle	C₂×C₃×C₈₁₁	D₄
hypercenter	C₈	D₂₄₃₃
hypocenter	C₃×C₈₁₁	D₈
LCS	G ▷ C₃×C₄×C₈₁₁ ▷ C₂×C₃×C₈₁₁ ▷ C₃×C₈₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 822
transitive degree	μ_t(G) = 19464

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

Polycyclic presentation:

2^4,3,811|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^810

Conjugacy classes & character table

9735 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, 811]);;
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.6^810);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D19464 #72274016