Finite Group Databasebeta

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Dic₄₀₅₂ #30633805

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

Splittings

Subgroups

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1029 points:

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

Polycyclic presentation:

2^4,1013|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^1012,c^b=cd,e^b=e^1012

Conjugacy classes & character table

4055 conjugacy classes. The character table is too large to display.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16208, 31);