D₉₅₆₄ #72214715

order	19128
automorphism	Aut(G) order 30451776
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₅₆₄

Splittings

Fitting	C₃×C₄×C₇₉₇ ⋊ C₂
Hall	C₃×C₇₉₇ ⋊ D₄
Sylow	C₇₉₇ ⋊ D₁₂
Sylow	C₃ ⋊ D₃₁₈₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₃₉₁
commutator, socle, cosocle	C₂×C₃×C₇₉₇	C₂²
Fitting	C₃×C₄×C₇₉₇	C₂
hypercenter	C₄	D₂₃₉₁
hypocenter	C₃×C₇₉₇	D₄
LCS	G ▷ C₂×C₃×C₇₉₇ ▷ C₃×C₇₉₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 804
transitive degree	μ_t(G) = 9564

A minimal faithful permutation representation — 2 generators on 804 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);(2,4)(6,7)(9,804)(10,803)(11,802)(12,801)(13,800)(14,799)(15,798)(16,797)(17,796)(18,795)(19,794)(20,793)(21,792)(22,791)(23,790)(24,789)(25,788)(26,787)(27,786)(28,785)(29,784)(30,783)(31,782)(32,781)(33,780)(34,779)(35,778)(36,777)(37,776)(38,775)(39,774)(40,773)(41,772)(42,771)(43,770)(44,769)(45,768)(46,767)(47,766)(48,765)(49,764)(50,763)(51,762)(52,761)(53,760)(54,759)(55,758)(56,757)(57,756)(58,755)(59,754)(60,753)(61,752)(62,751)(63,750)(64,749)(65,748)(66,747)(67,746)(68,745)(69,744)(70,743)(71,742)(72,741)(73,740)(74,739)(75,738)(76,737)(77,736)(78,735)(79,734)(80,733)(81,732)(82,731)(83,730)(84,729)(85,728)(86,727)(87,726)(88,725)(89,724)(90,723)(91,722)(92,721)(93,720)(94,719)(95,718)(96,717)(97,716)(98,715)(99,714)(100,713)(101,712)(102,711)(103,710)(104,709)(105,708)(106,707)(107,706)(108,705)(109,704)(110,703)(111,702)(112,701)(113,700)(114,699)(115,698)(116,697)(117,696)(118,695)(119,694)(120,693)(121,692)(122,691)(123,690)(124,689)(125,688)(126,687)(127,686)(128,685)(129,684)(130,683)(131,682)(132,681)(133,680)(134,679)(135,678)(136,677)(137,676)(138,675)(139,674)(140,673)(141,672)(142,671)(143,670)(144,669)(145,668)(146,667)(147,666)(148,665)(149,664)(150,663)(151,662)(152,661)(153,660)(154,659)(155,658)(156,657)(157,656)(158,655)(159,654)(160,653)(161,652)(162,651)(163,650)(164,649)(165,648)(166,647)(167,646)(168,645)(169,644)(170,643)(171,642)(172,641)(173,640)(174,639)(175,638)(176,637)(177,636)(178,635)(179,634)(180,633)(181,632)(182,631)(183,630)(184,629)(185,628)(186,627)(187,626)(188,625)(189,624)(190,623)(191,622)(192,621)(193,620)(194,619)(195,618)(196,617)(197,616)(198,615)(199,614)(200,613)(201,612)(202,611)(203,610)(204,609)(205,608)(206,607)(207,606)(208,605)(209,604)(210,603)(211,602)(212,601)(213,600)(214,599)(215,598)(216,597)(217,596)(218,595)(219,594)(220,593)(221,592)(222,591)(223,590)(224,589)(225,588)(226,587)(227,586)(228,585)(229,584)(230,583)(231,582)(232,581)(233,580)(234,579)(235,578)(236,577)(237,576)(238,575)(239,574)(240,573)(241,572)(242,571)(243,570)(244,569)(245,568)(246,567)(247,566)(248,565)(249,564)(250,563)(251,562)(252,561)(253,560)(254,559)(255,558)(256,557)(257,556)(258,555)(259,554)(260,553)(261,552)(262,551)(263,550)(264,549)(265,548)(266,547)(267,546)(268,545)(269,544)(270,543)(271,542)(272,541)(273,540)(274,539)(275,538)(276,537)(277,536)(278,535)(279,534)(280,533)(281,532)(282,531)(283,530)(284,529)(285,528)(286,527)(287,526)(288,525)(289,524)(290,523)(291,522)(292,521)(293,520)(294,519)(295,518)(296,517)(297,516)(298,515)(299,514)(300,513)(301,512)(302,511)(303,510)(304,509)(305,508)(306,507)(307,506)(308,505)(309,504)(310,503)(311,502)(312,501)(313,500)(314,499)(315,498)(316,497)(317,496)(318,495)(319,494)(320,493)(321,492)(322,491)(323,490)(324,489)(325,488)(326,487)(327,486)(328,485)(329,484)(330,483)(331,482)(332,481)(333,480)(334,479)(335,478)(336,477)(337,476)(338,475)(339,474)(340,473)(341,472)(342,471)(343,470)(344,469)(345,468)(346,467)(347,466)(348,465)(349,464)(350,463)(351,462)(352,461)(353,460)(354,459)(355,458)(356,457)(357,456)(358,455)(359,454)(360,453)(361,452)(362,451)(363,450)(364,449)(365,448)(366,447)(367,446)(368,445)(369,444)(370,443)(371,442)(372,441)(373,440)(374,439)(375,438)(376,437)(377,436)(378,435)(379,434)(380,433)(381,432)(382,431)(383,430)(384,429)(385,428)(386,427)(387,426)(388,425)(389,424)(390,423)(391,422)(392,421)(393,420)(394,419)(395,418)(396,417)(397,416)(398,415)(399,414)(400,413)(401,412)(402,411)(403,410)(404,409)(405,408)(406,407)

Polycyclic presentation:

2^3,3,797|b^2=c,b^a=bc,d^a=d^2,e^a=e^796

Conjugacy classes & character table

4785 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₉₁₂₈, D₂₈₆₉₂
maximal normal subgroup of	D₁₉₁₂₈
maximal quotient of	Dic₉₅₆₄, D₁₉₁₂₈, D₂₈₆₉₂

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 797]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^796);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9564 #72214715