D₂₀₈₇₁ #72274729

order	41742
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₀₈₇₁

Splittings

Fitting, derived, Hall	C₂₇×C₇₇₃ ⋊ C₂
Sylow	C₇₇₃ ⋊ D₂₇
Sylow	C₂₇ ⋊ D₇₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₇×C₇₇₃	C₂
Frattini	C₉	D₂₃₁₉
socle	C₃×C₇₇₃	D₉
LCS	G ▷ C₂₇×C₇₇₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 800
transitive degree	μ_t(G) = 20871

A minimal faithful permutation representation — 2 generators on 800 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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,800)(30,799)(31,798)(32,797)(33,796)(34,795)(35,794)(36,793)(37,792)(38,791)(39,790)(40,789)(41,788)(42,787)(43,786)(44,785)(45,784)(46,783)(47,782)(48,781)(49,780)(50,779)(51,778)(52,777)(53,776)(54,775)(55,774)(56,773)(57,772)(58,771)(59,770)(60,769)(61,768)(62,767)(63,766)(64,765)(65,764)(66,763)(67,762)(68,761)(69,760)(70,759)(71,758)(72,757)(73,756)(74,755)(75,754)(76,753)(77,752)(78,751)(79,750)(80,749)(81,748)(82,747)(83,746)(84,745)(85,744)(86,743)(87,742)(88,741)(89,740)(90,739)(91,738)(92,737)(93,736)(94,735)(95,734)(96,733)(97,732)(98,731)(99,730)(100,729)(101,728)(102,727)(103,726)(104,725)(105,724)(106,723)(107,722)(108,721)(109,720)(110,719)(111,718)(112,717)(113,716)(114,715)(115,714)(116,713)(117,712)(118,711)(119,710)(120,709)(121,708)(122,707)(123,706)(124,705)(125,704)(126,703)(127,702)(128,701)(129,700)(130,699)(131,698)(132,697)(133,696)(134,695)(135,694)(136,693)(137,692)(138,691)(139,690)(140,689)(141,688)(142,687)(143,686)(144,685)(145,684)(146,683)(147,682)(148,681)(149,680)(150,679)(151,678)(152,677)(153,676)(154,675)(155,674)(156,673)(157,672)(158,671)(159,670)(160,669)(161,668)(162,667)(163,666)(164,665)(165,664)(166,663)(167,662)(168,661)(169,660)(170,659)(171,658)(172,657)(173,656)(174,655)(175,654)(176,653)(177,652)(178,651)(179,650)(180,649)(181,648)(182,647)(183,646)(184,645)(185,644)(186,643)(187,642)(188,641)(189,640)(190,639)(191,638)(192,637)(193,636)(194,635)(195,634)(196,633)(197,632)(198,631)(199,630)(200,629)(201,628)(202,627)(203,626)(204,625)(205,624)(206,623)(207,622)(208,621)(209,620)(210,619)(211,618)(212,617)(213,616)(214,615)(215,614)(216,613)(217,612)(218,611)(219,610)(220,609)(221,608)(222,607)(223,606)(224,605)(225,604)(226,603)(227,602)(228,601)(229,600)(230,599)(231,598)(232,597)(233,596)(234,595)(235,594)(236,593)(237,592)(238,591)(239,590)(240,589)(241,588)(242,587)(243,586)(244,585)(245,584)(246,583)(247,582)(248,581)(249,580)(250,579)(251,578)(252,577)(253,576)(254,575)(255,574)(256,573)(257,572)(258,571)(259,570)(260,569)(261,568)(262,567)(263,566)(264,565)(265,564)(266,563)(267,562)(268,561)(269,560)(270,559)(271,558)(272,557)(273,556)(274,555)(275,554)(276,553)(277,552)(278,551)(279,550)(280,549)(281,548)(282,547)(283,546)(284,545)(285,544)(286,543)(287,542)(288,541)(289,540)(290,539)(291,538)(292,537)(293,536)(294,535)(295,534)(296,533)(297,532)(298,531)(299,530)(300,529)(301,528)(302,527)(303,526)(304,525)(305,524)(306,523)(307,522)(308,521)(309,520)(310,519)(311,518)(312,517)(313,516)(314,515)(315,514)(316,513)(317,512)(318,511)(319,510)(320,509)(321,508)(322,507)(323,506)(324,505)(325,504)(326,503)(327,502)(328,501)(329,500)(330,499)(331,498)(332,497)(333,496)(334,495)(335,494)(336,493)(337,492)(338,491)(339,490)(340,489)(341,488)(342,487)(343,486)(344,485)(345,484)(346,483)(347,482)(348,481)(349,480)(350,479)(351,478)(352,477)(353,476)(354,475)(355,474)(356,473)(357,472)(358,471)(359,470)(360,469)(361,468)(362,467)(363,466)(364,465)(365,464)(366,463)(367,462)(368,461)(369,460)(370,459)(371,458)(372,457)(373,456)(374,455)(375,454)(376,453)(377,452)(378,451)(379,450)(380,449)(381,448)(382,447)(383,446)(384,445)(385,444)(386,443)(387,442)(388,441)(389,440)(390,439)(391,438)(392,437)(393,436)(394,435)(395,434)(396,433)(397,432)(398,431)(399,430)(400,429)(401,428)(402,427)(403,426)(404,425)(405,424)(406,423)(407,422)(408,421)(409,420)(410,419)(411,418)(412,417)(413,416)(414,415)

Polycyclic presentation:

2,3,773,3^2|b^3=d,d^3=e,b^a=b^2d^2e^2,c^a=c^772,d^a=d^2e^2,e^a=e^2

Conjugacy classes & character table

10437 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");;
gap> coll := SingleCollector(F, [2, 3, 773, 3, 3]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.4^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^772);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.5^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D20871 #72274729