D₉₁₈₉ #32913604

order	18378
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₁₈₉
unambiguous semidirect product	C₉⋊D₁₀₂₁
unambiguous semidirect product	C₁₀₂₁⋊D₉

Splittings

Fitting, derived, Hall	C₉×C₁₀₂₁ ⋊ C₂
Sylow	C₁₀₂₁ ⋊ D₉
Sylow	C₉ ⋊ D₁₀₂₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₀₂₁	C₂
Frattini	C₃	D₃₀₆₃
socle	C₃×C₁₀₂₁	D₃
LCS	G ▷ C₉×C₁₀₂₁
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1030
transitive degree	μ_t(G) = 9189

A minimal faithful permutation representation — 2 generators on 1030 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030);(2,9)(3,8)(4,7)(5,6)(11,1030)(12,1029)(13,1028)(14,1027)(15,1026)(16,1025)(17,1024)(18,1023)(19,1022)(20,1021)(21,1020)(22,1019)(23,1018)(24,1017)(25,1016)(26,1015)(27,1014)(28,1013)(29,1012)(30,1011)(31,1010)(32,1009)(33,1008)(34,1007)(35,1006)(36,1005)(37,1004)(38,1003)(39,1002)(40,1001)(41,1000)(42,999)(43,998)(44,997)(45,996)(46,995)(47,994)(48,993)(49,992)(50,991)(51,990)(52,989)(53,988)(54,987)(55,986)(56,985)(57,984)(58,983)(59,982)(60,981)(61,980)(62,979)(63,978)(64,977)(65,976)(66,975)(67,974)(68,973)(69,972)(70,971)(71,970)(72,969)(73,968)(74,967)(75,966)(76,965)(77,964)(78,963)(79,962)(80,961)(81,960)(82,959)(83,958)(84,957)(85,956)(86,955)(87,954)(88,953)(89,952)(90,951)(91,950)(92,949)(93,948)(94,947)(95,946)(96,945)(97,944)(98,943)(99,942)(100,941)(101,940)(102,939)(103,938)(104,937)(105,936)(106,935)(107,934)(108,933)(109,932)(110,931)(111,930)(112,929)(113,928)(114,927)(115,926)(116,925)(117,924)(118,923)(119,922)(120,921)(121,920)(122,919)(123,918)(124,917)(125,916)(126,915)(127,914)(128,913)(129,912)(130,911)(131,910)(132,909)(133,908)(134,907)(135,906)(136,905)(137,904)(138,903)(139,902)(140,901)(141,900)(142,899)(143,898)(144,897)(145,896)(146,895)(147,894)(148,893)(149,892)(150,891)(151,890)(152,889)(153,888)(154,887)(155,886)(156,885)(157,884)(158,883)(159,882)(160,881)(161,880)(162,879)(163,878)(164,877)(165,876)(166,875)(167,874)(168,873)(169,872)(170,871)(171,870)(172,869)(173,868)(174,867)(175,866)(176,865)(177,864)(178,863)(179,862)(180,861)(181,860)(182,859)(183,858)(184,857)(185,856)(186,855)(187,854)(188,853)(189,852)(190,851)(191,850)(192,849)(193,848)(194,847)(195,846)(196,845)(197,844)(198,843)(199,842)(200,841)(201,840)(202,839)(203,838)(204,837)(205,836)(206,835)(207,834)(208,833)(209,832)(210,831)(211,830)(212,829)(213,828)(214,827)(215,826)(216,825)(217,824)(218,823)(219,822)(220,821)(221,820)(222,819)(223,818)(224,817)(225,816)(226,815)(227,814)(228,813)(229,812)(230,811)(231,810)(232,809)(233,808)(234,807)(235,806)(236,805)(237,804)(238,803)(239,802)(240,801)(241,800)(242,799)(243,798)(244,797)(245,796)(246,795)(247,794)(248,793)(249,792)(250,791)(251,790)(252,789)(253,788)(254,787)(255,786)(256,785)(257,784)(258,783)(259,782)(260,781)(261,780)(262,779)(263,778)(264,777)(265,776)(266,775)(267,774)(268,773)(269,772)(270,771)(271,770)(272,769)(273,768)(274,767)(275,766)(276,765)(277,764)(278,763)(279,762)(280,761)(281,760)(282,759)(283,758)(284,757)(285,756)(286,755)(287,754)(288,753)(289,752)(290,751)(291,750)(292,749)(293,748)(294,747)(295,746)(296,745)(297,744)(298,743)(299,742)(300,741)(301,740)(302,739)(303,738)(304,737)(305,736)(306,735)(307,734)(308,733)(309,732)(310,731)(311,730)(312,729)(313,728)(314,727)(315,726)(316,725)(317,724)(318,723)(319,722)(320,721)(321,720)(322,719)(323,718)(324,717)(325,716)(326,715)(327,714)(328,713)(329,712)(330,711)(331,710)(332,709)(333,708)(334,707)(335,706)(336,705)(337,704)(338,703)(339,702)(340,701)(341,700)(342,699)(343,698)(344,697)(345,696)(346,695)(347,694)(348,693)(349,692)(350,691)(351,690)(352,689)(353,688)(354,687)(355,686)(356,685)(357,684)(358,683)(359,682)(360,681)(361,680)(362,679)(363,678)(364,677)(365,676)(366,675)(367,674)(368,673)(369,672)(370,671)(371,670)(372,669)(373,668)(374,667)(375,666)(376,665)(377,664)(378,663)(379,662)(380,661)(381,660)(382,659)(383,658)(384,657)(385,656)(386,655)(387,654)(388,653)(389,652)(390,651)(391,650)(392,649)(393,648)(394,647)(395,646)(396,645)(397,644)(398,643)(399,642)(400,641)(401,640)(402,639)(403,638)(404,637)(405,636)(406,635)(407,634)(408,633)(409,632)(410,631)(411,630)(412,629)(413,628)(414,627)(415,626)(416,625)(417,624)(418,623)(419,622)(420,621)(421,620)(422,619)(423,618)(424,617)(425,616)(426,615)(427,614)(428,613)(429,612)(430,611)(431,610)(432,609)(433,608)(434,607)(435,606)(436,605)(437,604)(438,603)(439,602)(440,601)(441,600)(442,599)(443,598)(444,597)(445,596)(446,595)(447,594)(448,593)(449,592)(450,591)(451,590)(452,589)(453,588)(454,587)(455,586)(456,585)(457,584)(458,583)(459,582)(460,581)(461,580)(462,579)(463,578)(464,577)(465,576)(466,575)(467,574)(468,573)(469,572)(470,571)(471,570)(472,569)(473,568)(474,567)(475,566)(476,565)(477,564)(478,563)(479,562)(480,561)(481,560)(482,559)(483,558)(484,557)(485,556)(486,555)(487,554)(488,553)(489,552)(490,551)(491,550)(492,549)(493,548)(494,547)(495,546)(496,545)(497,544)(498,543)(499,542)(500,541)(501,540)(502,539)(503,538)(504,537)(505,536)(506,535)(507,534)(508,533)(509,532)(510,531)(511,530)(512,529)(513,528)(514,527)(515,526)(516,525)(517,524)(518,523)(519,522)(520,521)

Polycyclic presentation:

2,3,1021,3|b^3=d^2,b^a=b^2d,c^a=c^1020,d^a=d^2

Conjugacy classes & character table

4596 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉×D₁₀₂₁, C₂×D₉₁₈₉, D₂₇₅₆₇
maximal normal subgroup of	D₉×D₁₀₂₁, C₂×D₉₁₈₉
maximal quotient of	Dic₉₁₈₉, C₂×D₉₁₈₉, D₂₇₅₆₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18378, 6);

D9189 #32913604