Dic₁₀₀₉₀ #72274182

order	40360
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₀₉₀

Splittings

Hall	C₅×C₁₀₀₉ ⋊ Q₈
Sylow	C₁₀₀₉ ⋊ Dic₁₀
Sylow	C₅ ⋊ Dic₂₀₁₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₀₄₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₁₀₀₉	C₂²
Fitting	C₄×C₅×C₁₀₀₉	C₂
hypercenter	C₄	D₅₀₄₅
hypocenter	C₅×C₁₀₀₉	Q₈
LCS	G ▷ C₂×C₅×C₁₀₀₉ ▷ C₅×C₁₀₀₉
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1022
transitive degree	μ_t(G) = 40360

A minimal faithful permutation representation — 2 generators on 1022 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,1022)(16,1021)(17,1020)(18,1019)(19,1018)(20,1017)(21,1016)(22,1015)(23,1014)(24,1013)(25,1012)(26,1011)(27,1010)(28,1009)(29,1008)(30,1007)(31,1006)(32,1005)(33,1004)(34,1003)(35,1002)(36,1001)(37,1000)(38,999)(39,998)(40,997)(41,996)(42,995)(43,994)(44,993)(45,992)(46,991)(47,990)(48,989)(49,988)(50,987)(51,986)(52,985)(53,984)(54,983)(55,982)(56,981)(57,980)(58,979)(59,978)(60,977)(61,976)(62,975)(63,974)(64,973)(65,972)(66,971)(67,970)(68,969)(69,968)(70,967)(71,966)(72,965)(73,964)(74,963)(75,962)(76,961)(77,960)(78,959)(79,958)(80,957)(81,956)(82,955)(83,954)(84,953)(85,952)(86,951)(87,950)(88,949)(89,948)(90,947)(91,946)(92,945)(93,944)(94,943)(95,942)(96,941)(97,940)(98,939)(99,938)(100,937)(101,936)(102,935)(103,934)(104,933)(105,932)(106,931)(107,930)(108,929)(109,928)(110,927)(111,926)(112,925)(113,924)(114,923)(115,922)(116,921)(117,920)(118,919)(119,918)(120,917)(121,916)(122,915)(123,914)(124,913)(125,912)(126,911)(127,910)(128,909)(129,908)(130,907)(131,906)(132,905)(133,904)(134,903)(135,902)(136,901)(137,900)(138,899)(139,898)(140,897)(141,896)(142,895)(143,894)(144,893)(145,892)(146,891)(147,890)(148,889)(149,888)(150,887)(151,886)(152,885)(153,884)(154,883)(155,882)(156,881)(157,880)(158,879)(159,878)(160,877)(161,876)(162,875)(163,874)(164,873)(165,872)(166,871)(167,870)(168,869)(169,868)(170,867)(171,866)(172,865)(173,864)(174,863)(175,862)(176,861)(177,860)(178,859)(179,858)(180,857)(181,856)(182,855)(183,854)(184,853)(185,852)(186,851)(187,850)(188,849)(189,848)(190,847)(191,846)(192,845)(193,844)(194,843)(195,842)(196,841)(197,840)(198,839)(199,838)(200,837)(201,836)(202,835)(203,834)(204,833)(205,832)(206,831)(207,830)(208,829)(209,828)(210,827)(211,826)(212,825)(213,824)(214,823)(215,822)(216,821)(217,820)(218,819)(219,818)(220,817)(221,816)(222,815)(223,814)(224,813)(225,812)(226,811)(227,810)(228,809)(229,808)(230,807)(231,806)(232,805)(233,804)(234,803)(235,802)(236,801)(237,800)(238,799)(239,798)(240,797)(241,796)(242,795)(243,794)(244,793)(245,792)(246,791)(247,790)(248,789)(249,788)(250,787)(251,786)(252,785)(253,784)(254,783)(255,782)(256,781)(257,780)(258,779)(259,778)(260,777)(261,776)(262,775)(263,774)(264,773)(265,772)(266,771)(267,770)(268,769)(269,768)(270,767)(271,766)(272,765)(273,764)(274,763)(275,762)(276,761)(277,760)(278,759)(279,758)(280,757)(281,756)(282,755)(283,754)(284,753)(285,752)(286,751)(287,750)(288,749)(289,748)(290,747)(291,746)(292,745)(293,744)(294,743)(295,742)(296,741)(297,740)(298,739)(299,738)(300,737)(301,736)(302,735)(303,734)(304,733)(305,732)(306,731)(307,730)(308,729)(309,728)(310,727)(311,726)(312,725)(313,724)(314,723)(315,722)(316,721)(317,720)(318,719)(319,718)(320,717)(321,716)(322,715)(323,714)(324,713)(325,712)(326,711)(327,710)(328,709)(329,708)(330,707)(331,706)(332,705)(333,704)(334,703)(335,702)(336,701)(337,700)(338,699)(339,698)(340,697)(341,696)(342,695)(343,694)(344,693)(345,692)(346,691)(347,690)(348,689)(349,688)(350,687)(351,686)(352,685)(353,684)(354,683)(355,682)(356,681)(357,680)(358,679)(359,678)(360,677)(361,676)(362,675)(363,674)(364,673)(365,672)(366,671)(367,670)(368,669)(369,668)(370,667)(371,666)(372,665)(373,664)(374,663)(375,662)(376,661)(377,660)(378,659)(379,658)(380,657)(381,656)(382,655)(383,654)(384,653)(385,652)(386,651)(387,650)(388,649)(389,648)(390,647)(391,646)(392,645)(393,644)(394,643)(395,642)(396,641)(397,640)(398,639)(399,638)(400,637)(401,636)(402,635)(403,634)(404,633)(405,632)(406,631)(407,630)(408,629)(409,628)(410,627)(411,626)(412,625)(413,624)(414,623)(415,622)(416,621)(417,620)(418,619)(419,618)(420,617)(421,616)(422,615)(423,614)(424,613)(425,612)(426,611)(427,610)(428,609)(429,608)(430,607)(431,606)(432,605)(433,604)(434,603)(435,602)(436,601)(437,600)(438,599)(439,598)(440,597)(441,596)(442,595)(443,594)(444,593)(445,592)(446,591)(447,590)(448,589)(449,588)(450,587)(451,586)(452,585)(453,584)(454,583)(455,582)(456,581)(457,580)(458,579)(459,578)(460,577)(461,576)(462,575)(463,574)(464,573)(465,572)(466,571)(467,570)(468,569)(469,568)(470,567)(471,566)(472,565)(473,564)(474,563)(475,562)(476,561)(477,560)(478,559)(479,558)(480,557)(481,556)(482,555)(483,554)(484,553)(485,552)(486,551)(487,550)(488,549)(489,548)(490,547)(491,546)(492,545)(493,544)(494,543)(495,542)(496,541)(497,540)(498,539)(499,538)(500,537)(501,536)(502,535)(503,534)(504,533)(505,532)(506,531)(507,530)(508,529)(509,528)(510,527)(511,526)(512,525)(513,524)(514,523)(515,522)(516,521)(517,520)(518,519)

Polycyclic presentation:

2^3,5,1009|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^1008

Conjugacy classes & character table

10093 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, 1009]);;
gap> SetPower(coll, 1, F.3);;
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.5^1008);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic10090 #72274182