D₉₁₇₁ #32913372

order	18342
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₁₇₁
unambiguous semidirect product	C₉⋊D₁₀₁₉
unambiguous semidirect product	C₁₀₁₉⋊D₉

Splittings

Fitting, derived, Hall	C₉×C₁₀₁₉ ⋊ C₂
Sylow	C₁₀₁₉ ⋊ D₉
Sylow	C₉ ⋊ D₁₀₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₀₁₉	C₂
Frattini	C₃	D₃₀₅₇
socle	C₃×C₁₀₁₉	D₃
LCS	G ▷ C₉×C₁₀₁₉
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1028
transitive degree	μ_t(G) = 9171

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

Polycyclic presentation:

2,3,1019,3|b^3=d^2,b^a=b^2d,c^a=c^1018,d^a=d^2

Conjugacy classes & character table

4587 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉×D₁₀₁₉, C₂×D₉₁₇₁, D₂₇₅₁₃
maximal normal subgroup of	D₉×D₁₀₁₉, C₂×D₉₁₇₁
maximal quotient of	Dic₉₁₇₁, C₂×D₉₁₇₁, D₂₇₅₁₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18342, 4);

D9171 #32913372