D₉₀₈₁ #31827795

order	18162
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₀₈₁
unambiguous semidirect product	C₉⋊D₁₀₀₉
unambiguous semidirect product	C₁₀₀₉⋊D₉

Splittings

Fitting, derived, Hall	C₉×C₁₀₀₉ ⋊ C₂
Sylow	C₁₀₀₉ ⋊ D₉
Sylow	C₉ ⋊ D₁₀₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₀₀₉	C₂
Frattini	C₃	D₃₀₂₇
socle	C₃×C₁₀₀₉	D₃
LCS	G ▷ C₉×C₁₀₀₉
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1018
transitive degree	μ_t(G) = 9081

A minimal faithful permutation representation — 2 generators on 1018 points:

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

Polycyclic presentation:

2,3,1009,3|b^3=d^2,b^a=b^2d,c^a=c^1008,d^a=d^2

Conjugacy classes & character table

4542 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉×D₁₀₀₉, C₂×D₉₀₈₁, D₂₇₂₄₃
maximal normal subgroup of	D₉×D₁₀₀₉, C₂×D₉₀₈₁
maximal quotient of	Dic₉₀₈₁, C₂×D₉₀₈₁, D₂₇₂₄₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18162, 6);

D9081 #31827795