D₉₃₅₁ #34035286

order	18702
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₃₅₁
unambiguous semidirect product	C₉⋊D₁₀₃₉
unambiguous semidirect product	C₁₀₃₉⋊D₉

Splittings

Fitting, derived, Hall	C₉×C₁₀₃₉ ⋊ C₂
Sylow	C₁₀₃₉ ⋊ D₉
Sylow	C₉ ⋊ D₁₀₃₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₀₃₉	C₂
Frattini	C₃	D₃₁₁₇
socle	C₃×C₁₀₃₉	D₃
LCS	G ▷ C₉×C₁₀₃₉
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1048
transitive degree	μ_t(G) = 9351

A minimal faithful permutation representation — 2 generators on 1048 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048);(2,9)(3,8)(4,7)(5,6)(11,1048)(12,1047)(13,1046)(14,1045)(15,1044)(16,1043)(17,1042)(18,1041)(19,1040)(20,1039)(21,1038)(22,1037)(23,1036)(24,1035)(25,1034)(26,1033)(27,1032)(28,1031)(29,1030)(30,1029)(31,1028)(32,1027)(33,1026)(34,1025)(35,1024)(36,1023)(37,1022)(38,1021)(39,1020)(40,1019)(41,1018)(42,1017)(43,1016)(44,1015)(45,1014)(46,1013)(47,1012)(48,1011)(49,1010)(50,1009)(51,1008)(52,1007)(53,1006)(54,1005)(55,1004)(56,1003)(57,1002)(58,1001)(59,1000)(60,999)(61,998)(62,997)(63,996)(64,995)(65,994)(66,993)(67,992)(68,991)(69,990)(70,989)(71,988)(72,987)(73,986)(74,985)(75,984)(76,983)(77,982)(78,981)(79,980)(80,979)(81,978)(82,977)(83,976)(84,975)(85,974)(86,973)(87,972)(88,971)(89,970)(90,969)(91,968)(92,967)(93,966)(94,965)(95,964)(96,963)(97,962)(98,961)(99,960)(100,959)(101,958)(102,957)(103,956)(104,955)(105,954)(106,953)(107,952)(108,951)(109,950)(110,949)(111,948)(112,947)(113,946)(114,945)(115,944)(116,943)(117,942)(118,941)(119,940)(120,939)(121,938)(122,937)(123,936)(124,935)(125,934)(126,933)(127,932)(128,931)(129,930)(130,929)(131,928)(132,927)(133,926)(134,925)(135,924)(136,923)(137,922)(138,921)(139,920)(140,919)(141,918)(142,917)(143,916)(144,915)(145,914)(146,913)(147,912)(148,911)(149,910)(150,909)(151,908)(152,907)(153,906)(154,905)(155,904)(156,903)(157,902)(158,901)(159,900)(160,899)(161,898)(162,897)(163,896)(164,895)(165,894)(166,893)(167,892)(168,891)(169,890)(170,889)(171,888)(172,887)(173,886)(174,885)(175,884)(176,883)(177,882)(178,881)(179,880)(180,879)(181,878)(182,877)(183,876)(184,875)(185,874)(186,873)(187,872)(188,871)(189,870)(190,869)(191,868)(192,867)(193,866)(194,865)(195,864)(196,863)(197,862)(198,861)(199,860)(200,859)(201,858)(202,857)(203,856)(204,855)(205,854)(206,853)(207,852)(208,851)(209,850)(210,849)(211,848)(212,847)(213,846)(214,845)(215,844)(216,843)(217,842)(218,841)(219,840)(220,839)(221,838)(222,837)(223,836)(224,835)(225,834)(226,833)(227,832)(228,831)(229,830)(230,829)(231,828)(232,827)(233,826)(234,825)(235,824)(236,823)(237,822)(238,821)(239,820)(240,819)(241,818)(242,817)(243,816)(244,815)(245,814)(246,813)(247,812)(248,811)(249,810)(250,809)(251,808)(252,807)(253,806)(254,805)(255,804)(256,803)(257,802)(258,801)(259,800)(260,799)(261,798)(262,797)(263,796)(264,795)(265,794)(266,793)(267,792)(268,791)(269,790)(270,789)(271,788)(272,787)(273,786)(274,785)(275,784)(276,783)(277,782)(278,781)(279,780)(280,779)(281,778)(282,777)(283,776)(284,775)(285,774)(286,773)(287,772)(288,771)(289,770)(290,769)(291,768)(292,767)(293,766)(294,765)(295,764)(296,763)(297,762)(298,761)(299,760)(300,759)(301,758)(302,757)(303,756)(304,755)(305,754)(306,753)(307,752)(308,751)(309,750)(310,749)(311,748)(312,747)(313,746)(314,745)(315,744)(316,743)(317,742)(318,741)(319,740)(320,739)(321,738)(322,737)(323,736)(324,735)(325,734)(326,733)(327,732)(328,731)(329,730)(330,729)(331,728)(332,727)(333,726)(334,725)(335,724)(336,723)(337,722)(338,721)(339,720)(340,719)(341,718)(342,717)(343,716)(344,715)(345,714)(346,713)(347,712)(348,711)(349,710)(350,709)(351,708)(352,707)(353,706)(354,705)(355,704)(356,703)(357,702)(358,701)(359,700)(360,699)(361,698)(362,697)(363,696)(364,695)(365,694)(366,693)(367,692)(368,691)(369,690)(370,689)(371,688)(372,687)(373,686)(374,685)(375,684)(376,683)(377,682)(378,681)(379,680)(380,679)(381,678)(382,677)(383,676)(384,675)(385,674)(386,673)(387,672)(388,671)(389,670)(390,669)(391,668)(392,667)(393,666)(394,665)(395,664)(396,663)(397,662)(398,661)(399,660)(400,659)(401,658)(402,657)(403,656)(404,655)(405,654)(406,653)(407,652)(408,651)(409,650)(410,649)(411,648)(412,647)(413,646)(414,645)(415,644)(416,643)(417,642)(418,641)(419,640)(420,639)(421,638)(422,637)(423,636)(424,635)(425,634)(426,633)(427,632)(428,631)(429,630)(430,629)(431,628)(432,627)(433,626)(434,625)(435,624)(436,623)(437,622)(438,621)(439,620)(440,619)(441,618)(442,617)(443,616)(444,615)(445,614)(446,613)(447,612)(448,611)(449,610)(450,609)(451,608)(452,607)(453,606)(454,605)(455,604)(456,603)(457,602)(458,601)(459,600)(460,599)(461,598)(462,597)(463,596)(464,595)(465,594)(466,593)(467,592)(468,591)(469,590)(470,589)(471,588)(472,587)(473,586)(474,585)(475,584)(476,583)(477,582)(478,581)(479,580)(480,579)(481,578)(482,577)(483,576)(484,575)(485,574)(486,573)(487,572)(488,571)(489,570)(490,569)(491,568)(492,567)(493,566)(494,565)(495,564)(496,563)(497,562)(498,561)(499,560)(500,559)(501,558)(502,557)(503,556)(504,555)(505,554)(506,553)(507,552)(508,551)(509,550)(510,549)(511,548)(512,547)(513,546)(514,545)(515,544)(516,543)(517,542)(518,541)(519,540)(520,539)(521,538)(522,537)(523,536)(524,535)(525,534)(526,533)(527,532)(528,531)(529,530)

Polycyclic presentation:

2,3,1039,3|b^3=d^2,b^a=b^2d,c^a=c^1038,d^a=d^2

Conjugacy classes & character table

4677 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉×D₁₀₃₉, C₂×D₉₃₅₁, D₂₈₀₅₃
maximal normal subgroup of	D₉×D₁₀₃₉, C₂×D₉₃₅₁
maximal quotient of	Dic₉₃₅₁, C₂×D₉₃₅₁, D₂₈₀₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18702, 6);

D9351 #34035286