Finite Group Databasebeta

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D₁₅₇₂₈ #44537905

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

Polycyclic presentation:

2^5,983|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^982,c^b=cd,d^b=de,f^b=f^982

Conjugacy classes & character table

7867 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31456, 88);