Finite Group Databasebeta

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Dic₈₀₇₂ #44567816

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

Aliases

Splittings

Subgroups

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 1041 points:

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

Polycyclic presentation:

2^5,1009|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^1008,c^b=cd,d^b=de,f^b=f^1008

Conjugacy classes & character table

8075 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32288, 93);