Finite Group Databasebeta

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D₁₅₈₅₆ #44540392

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

Aliases

Splittings

Subgroups

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

Polycyclic presentation:

2^5,991|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^990,c^b=cd,d^b=de,f^b=f^990

Conjugacy classes & character table

7931 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31712, 88);