Finite Group Databasebeta

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D₁₀₀₉ #14523071

• metacyclic
• supersolvable
• metabelian
• solvable
• monolithic
• monomial
• stem
• Frobenius
• ambivalent

Aliases

Splittings

Subgroups

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

Polycyclic presentation:

2,1009|b^a=b^1008

Conjugacy classes & character table

506 conjugacy classes — view the full character table.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2018, 1);