D₁₈₃₈₀ #72273540

order	36760
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₈₃₈₀

Splittings

Fitting	C₄×C₅×C₉₁₉ ⋊ C₂
Hall	C₅×C₉₁₉ ⋊ D₄
Sylow	C₉₁₉ ⋊ D₂₀
Sylow	C₅ ⋊ D₃₆₇₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₄₅₉₅
commutator, socle, cosocle	C₂×C₅×C₉₁₉	C₂²
Fitting	C₄×C₅×C₉₁₉	C₂
hypercenter	C₄	D₄₅₉₅
hypocenter	C₅×C₉₁₉	D₄
LCS	G ▷ C₂×C₅×C₉₁₉ ▷ C₅×C₉₁₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 928
transitive degree	μ_t(G) = 18380

A minimal faithful permutation representation — 2 generators on 928 points:

(1,2,3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928);(2,4)(6,9)(7,8)(11,928)(12,927)(13,926)(14,925)(15,924)(16,923)(17,922)(18,921)(19,920)(20,919)(21,918)(22,917)(23,916)(24,915)(25,914)(26,913)(27,912)(28,911)(29,910)(30,909)(31,908)(32,907)(33,906)(34,905)(35,904)(36,903)(37,902)(38,901)(39,900)(40,899)(41,898)(42,897)(43,896)(44,895)(45,894)(46,893)(47,892)(48,891)(49,890)(50,889)(51,888)(52,887)(53,886)(54,885)(55,884)(56,883)(57,882)(58,881)(59,880)(60,879)(61,878)(62,877)(63,876)(64,875)(65,874)(66,873)(67,872)(68,871)(69,870)(70,869)(71,868)(72,867)(73,866)(74,865)(75,864)(76,863)(77,862)(78,861)(79,860)(80,859)(81,858)(82,857)(83,856)(84,855)(85,854)(86,853)(87,852)(88,851)(89,850)(90,849)(91,848)(92,847)(93,846)(94,845)(95,844)(96,843)(97,842)(98,841)(99,840)(100,839)(101,838)(102,837)(103,836)(104,835)(105,834)(106,833)(107,832)(108,831)(109,830)(110,829)(111,828)(112,827)(113,826)(114,825)(115,824)(116,823)(117,822)(118,821)(119,820)(120,819)(121,818)(122,817)(123,816)(124,815)(125,814)(126,813)(127,812)(128,811)(129,810)(130,809)(131,808)(132,807)(133,806)(134,805)(135,804)(136,803)(137,802)(138,801)(139,800)(140,799)(141,798)(142,797)(143,796)(144,795)(145,794)(146,793)(147,792)(148,791)(149,790)(150,789)(151,788)(152,787)(153,786)(154,785)(155,784)(156,783)(157,782)(158,781)(159,780)(160,779)(161,778)(162,777)(163,776)(164,775)(165,774)(166,773)(167,772)(168,771)(169,770)(170,769)(171,768)(172,767)(173,766)(174,765)(175,764)(176,763)(177,762)(178,761)(179,760)(180,759)(181,758)(182,757)(183,756)(184,755)(185,754)(186,753)(187,752)(188,751)(189,750)(190,749)(191,748)(192,747)(193,746)(194,745)(195,744)(196,743)(197,742)(198,741)(199,740)(200,739)(201,738)(202,737)(203,736)(204,735)(205,734)(206,733)(207,732)(208,731)(209,730)(210,729)(211,728)(212,727)(213,726)(214,725)(215,724)(216,723)(217,722)(218,721)(219,720)(220,719)(221,718)(222,717)(223,716)(224,715)(225,714)(226,713)(227,712)(228,711)(229,710)(230,709)(231,708)(232,707)(233,706)(234,705)(235,704)(236,703)(237,702)(238,701)(239,700)(240,699)(241,698)(242,697)(243,696)(244,695)(245,694)(246,693)(247,692)(248,691)(249,690)(250,689)(251,688)(252,687)(253,686)(254,685)(255,684)(256,683)(257,682)(258,681)(259,680)(260,679)(261,678)(262,677)(263,676)(264,675)(265,674)(266,673)(267,672)(268,671)(269,670)(270,669)(271,668)(272,667)(273,666)(274,665)(275,664)(276,663)(277,662)(278,661)(279,660)(280,659)(281,658)(282,657)(283,656)(284,655)(285,654)(286,653)(287,652)(288,651)(289,650)(290,649)(291,648)(292,647)(293,646)(294,645)(295,644)(296,643)(297,642)(298,641)(299,640)(300,639)(301,638)(302,637)(303,636)(304,635)(305,634)(306,633)(307,632)(308,631)(309,630)(310,629)(311,628)(312,627)(313,626)(314,625)(315,624)(316,623)(317,622)(318,621)(319,620)(320,619)(321,618)(322,617)(323,616)(324,615)(325,614)(326,613)(327,612)(328,611)(329,610)(330,609)(331,608)(332,607)(333,606)(334,605)(335,604)(336,603)(337,602)(338,601)(339,600)(340,599)(341,598)(342,597)(343,596)(344,595)(345,594)(346,593)(347,592)(348,591)(349,590)(350,589)(351,588)(352,587)(353,586)(354,585)(355,584)(356,583)(357,582)(358,581)(359,580)(360,579)(361,578)(362,577)(363,576)(364,575)(365,574)(366,573)(367,572)(368,571)(369,570)(370,569)(371,568)(372,567)(373,566)(374,565)(375,564)(376,563)(377,562)(378,561)(379,560)(380,559)(381,558)(382,557)(383,556)(384,555)(385,554)(386,553)(387,552)(388,551)(389,550)(390,549)(391,548)(392,547)(393,546)(394,545)(395,544)(396,543)(397,542)(398,541)(399,540)(400,539)(401,538)(402,537)(403,536)(404,535)(405,534)(406,533)(407,532)(408,531)(409,530)(410,529)(411,528)(412,527)(413,526)(414,525)(415,524)(416,523)(417,522)(418,521)(419,520)(420,519)(421,518)(422,517)(423,516)(424,515)(425,514)(426,513)(427,512)(428,511)(429,510)(430,509)(431,508)(432,507)(433,506)(434,505)(435,504)(436,503)(437,502)(438,501)(439,500)(440,499)(441,498)(442,497)(443,496)(444,495)(445,494)(446,493)(447,492)(448,491)(449,490)(450,489)(451,488)(452,487)(453,486)(454,485)(455,484)(456,483)(457,482)(458,481)(459,480)(460,479)(461,478)(462,477)(463,476)(464,475)(465,474)(466,473)(467,472)(468,471)(469,470)

Polycyclic presentation:

2^3,5,919|b^2=c,b^a=bc,d^a=d^4,e^a=e^918

Conjugacy classes & character table

9193 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, 2, 2, 5, 919]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^918);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D18380 #72273540