D₉₂₉₇ #32915787

order	18594
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₂₉₇
unambiguous semidirect product	C₉⋊D₁₀₃₃
unambiguous semidirect product	C₁₀₃₃⋊D₉

Splittings

Fitting, derived, Hall	C₉×C₁₀₃₃ ⋊ C₂
Sylow	C₁₀₃₃ ⋊ D₉
Sylow	C₉ ⋊ D₁₀₃₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₀₃₃	C₂
Frattini	C₃	D₃₀₉₉
socle	C₃×C₁₀₃₃	D₃
LCS	G ▷ C₉×C₁₀₃₃
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1042
transitive degree	μ_t(G) = 9297

A minimal faithful permutation representation — 2 generators on 1042 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042);(2,9)(3,8)(4,7)(5,6)(11,1042)(12,1041)(13,1040)(14,1039)(15,1038)(16,1037)(17,1036)(18,1035)(19,1034)(20,1033)(21,1032)(22,1031)(23,1030)(24,1029)(25,1028)(26,1027)(27,1026)(28,1025)(29,1024)(30,1023)(31,1022)(32,1021)(33,1020)(34,1019)(35,1018)(36,1017)(37,1016)(38,1015)(39,1014)(40,1013)(41,1012)(42,1011)(43,1010)(44,1009)(45,1008)(46,1007)(47,1006)(48,1005)(49,1004)(50,1003)(51,1002)(52,1001)(53,1000)(54,999)(55,998)(56,997)(57,996)(58,995)(59,994)(60,993)(61,992)(62,991)(63,990)(64,989)(65,988)(66,987)(67,986)(68,985)(69,984)(70,983)(71,982)(72,981)(73,980)(74,979)(75,978)(76,977)(77,976)(78,975)(79,974)(80,973)(81,972)(82,971)(83,970)(84,969)(85,968)(86,967)(87,966)(88,965)(89,964)(90,963)(91,962)(92,961)(93,960)(94,959)(95,958)(96,957)(97,956)(98,955)(99,954)(100,953)(101,952)(102,951)(103,950)(104,949)(105,948)(106,947)(107,946)(108,945)(109,944)(110,943)(111,942)(112,941)(113,940)(114,939)(115,938)(116,937)(117,936)(118,935)(119,934)(120,933)(121,932)(122,931)(123,930)(124,929)(125,928)(126,927)(127,926)(128,925)(129,924)(130,923)(131,922)(132,921)(133,920)(134,919)(135,918)(136,917)(137,916)(138,915)(139,914)(140,913)(141,912)(142,911)(143,910)(144,909)(145,908)(146,907)(147,906)(148,905)(149,904)(150,903)(151,902)(152,901)(153,900)(154,899)(155,898)(156,897)(157,896)(158,895)(159,894)(160,893)(161,892)(162,891)(163,890)(164,889)(165,888)(166,887)(167,886)(168,885)(169,884)(170,883)(171,882)(172,881)(173,880)(174,879)(175,878)(176,877)(177,876)(178,875)(179,874)(180,873)(181,872)(182,871)(183,870)(184,869)(185,868)(186,867)(187,866)(188,865)(189,864)(190,863)(191,862)(192,861)(193,860)(194,859)(195,858)(196,857)(197,856)(198,855)(199,854)(200,853)(201,852)(202,851)(203,850)(204,849)(205,848)(206,847)(207,846)(208,845)(209,844)(210,843)(211,842)(212,841)(213,840)(214,839)(215,838)(216,837)(217,836)(218,835)(219,834)(220,833)(221,832)(222,831)(223,830)(224,829)(225,828)(226,827)(227,826)(228,825)(229,824)(230,823)(231,822)(232,821)(233,820)(234,819)(235,818)(236,817)(237,816)(238,815)(239,814)(240,813)(241,812)(242,811)(243,810)(244,809)(245,808)(246,807)(247,806)(248,805)(249,804)(250,803)(251,802)(252,801)(253,800)(254,799)(255,798)(256,797)(257,796)(258,795)(259,794)(260,793)(261,792)(262,791)(263,790)(264,789)(265,788)(266,787)(267,786)(268,785)(269,784)(270,783)(271,782)(272,781)(273,780)(274,779)(275,778)(276,777)(277,776)(278,775)(279,774)(280,773)(281,772)(282,771)(283,770)(284,769)(285,768)(286,767)(287,766)(288,765)(289,764)(290,763)(291,762)(292,761)(293,760)(294,759)(295,758)(296,757)(297,756)(298,755)(299,754)(300,753)(301,752)(302,751)(303,750)(304,749)(305,748)(306,747)(307,746)(308,745)(309,744)(310,743)(311,742)(312,741)(313,740)(314,739)(315,738)(316,737)(317,736)(318,735)(319,734)(320,733)(321,732)(322,731)(323,730)(324,729)(325,728)(326,727)(327,726)(328,725)(329,724)(330,723)(331,722)(332,721)(333,720)(334,719)(335,718)(336,717)(337,716)(338,715)(339,714)(340,713)(341,712)(342,711)(343,710)(344,709)(345,708)(346,707)(347,706)(348,705)(349,704)(350,703)(351,702)(352,701)(353,700)(354,699)(355,698)(356,697)(357,696)(358,695)(359,694)(360,693)(361,692)(362,691)(363,690)(364,689)(365,688)(366,687)(367,686)(368,685)(369,684)(370,683)(371,682)(372,681)(373,680)(374,679)(375,678)(376,677)(377,676)(378,675)(379,674)(380,673)(381,672)(382,671)(383,670)(384,669)(385,668)(386,667)(387,666)(388,665)(389,664)(390,663)(391,662)(392,661)(393,660)(394,659)(395,658)(396,657)(397,656)(398,655)(399,654)(400,653)(401,652)(402,651)(403,650)(404,649)(405,648)(406,647)(407,646)(408,645)(409,644)(410,643)(411,642)(412,641)(413,640)(414,639)(415,638)(416,637)(417,636)(418,635)(419,634)(420,633)(421,632)(422,631)(423,630)(424,629)(425,628)(426,627)(427,626)(428,625)(429,624)(430,623)(431,622)(432,621)(433,620)(434,619)(435,618)(436,617)(437,616)(438,615)(439,614)(440,613)(441,612)(442,611)(443,610)(444,609)(445,608)(446,607)(447,606)(448,605)(449,604)(450,603)(451,602)(452,601)(453,600)(454,599)(455,598)(456,597)(457,596)(458,595)(459,594)(460,593)(461,592)(462,591)(463,590)(464,589)(465,588)(466,587)(467,586)(468,585)(469,584)(470,583)(471,582)(472,581)(473,580)(474,579)(475,578)(476,577)(477,576)(478,575)(479,574)(480,573)(481,572)(482,571)(483,570)(484,569)(485,568)(486,567)(487,566)(488,565)(489,564)(490,563)(491,562)(492,561)(493,560)(494,559)(495,558)(496,557)(497,556)(498,555)(499,554)(500,553)(501,552)(502,551)(503,550)(504,549)(505,548)(506,547)(507,546)(508,545)(509,544)(510,543)(511,542)(512,541)(513,540)(514,539)(515,538)(516,537)(517,536)(518,535)(519,534)(520,533)(521,532)(522,531)(523,530)(524,529)(525,528)(526,527)

Polycyclic presentation:

2,3,1033,3|b^3=d^2,b^a=b^2d,c^a=c^1032,d^a=d^2

Conjugacy classes & character table

4650 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉×D₁₀₃₃, C₂×D₉₂₉₇, D₂₇₈₉₁
maximal normal subgroup of	D₉×D₁₀₃₃, C₂×D₉₂₉₇
maximal quotient of	Dic₉₂₉₇, C₂×D₉₂₉₇, D₂₇₈₉₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18594, 6);

D9297 #32915787