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Dic5034 #72216617
Aliases
Splittings
Subgroups
24 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 850 points:
(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,11)(13,850)(14,849)(15,848)(16,847)(17,846)(18,845)(19,844)(20,843)(21,842)(22,841)(23,840)(24,839)(25,838)(26,837)(27,836)(28,835)(29,834)(30,833)(31,832)(32,831)(33,830)(34,829)(35,828)(36,827)(37,826)(38,825)(39,824)(40,823)(41,822)(42,821)(43,820)(44,819)(45,818)(46,817)(47,816)(48,815)(49,814)(50,813)(51,812)(52,811)(53,810)(54,809)(55,808)(56,807)(57,806)(58,805)(59,804)(60,803)(61,802)(62,801)(63,800)(64,799)(65,798)(66,797)(67,796)(68,795)(69,794)(70,793)(71,792)(72,791)(73,790)(74,789)(75,788)(76,787)(77,786)(78,785)(79,784)(80,783)(81,782)(82,781)(83,780)(84,779)(85,778)(86,777)(87,776)(88,775)(89,774)(90,773)(91,772)(92,771)(93,770)(94,769)(95,768)(96,767)(97,766)(98,765)(99,764)(100,763)(101,762)(102,761)(103,760)(104,759)(105,758)(106,757)(107,756)(108,755)(109,754)(110,753)(111,752)(112,751)(113,750)(114,749)(115,748)(116,747)(117,746)(118,745)(119,744)(120,743)(121,742)(122,741)(123,740)(124,739)(125,738)(126,737)(127,736)(128,735)(129,734)(130,733)(131,732)(132,731)(133,730)(134,729)(135,728)(136,727)(137,726)(138,725)(139,724)(140,723)(141,722)(142,721)(143,720)(144,719)(145,718)(146,717)(147,716)(148,715)(149,714)(150,713)(151,712)(152,711)(153,710)(154,709)(155,708)(156,707)(157,706)(158,705)(159,704)(160,703)(161,702)(162,701)(163,700)(164,699)(165,698)(166,697)(167,696)(168,695)(169,694)(170,693)(171,692)(172,691)(173,690)(174,689)(175,688)(176,687)(177,686)(178,685)(179,684)(180,683)(181,682)(182,681)(183,680)(184,679)(185,678)(186,677)(187,676)(188,675)(189,674)(190,673)(191,672)(192,671)(193,670)(194,669)(195,668)(196,667)(197,666)(198,665)(199,664)(200,663)(201,662)(202,661)(203,660)(204,659)(205,658)(206,657)(207,656)(208,655)(209,654)(210,653)(211,652)(212,651)(213,650)(214,649)(215,648)(216,647)(217,646)(218,645)(219,644)(220,643)(221,642)(222,641)(223,640)(224,639)(225,638)(226,637)(227,636)(228,635)(229,634)(230,633)(231,632)(232,631)(233,630)(234,629)(235,628)(236,627)(237,626)(238,625)(239,624)(240,623)(241,622)(242,621)(243,620)(244,619)(245,618)(246,617)(247,616)(248,615)(249,614)(250,613)(251,612)(252,611)(253,610)(254,609)(255,608)(256,607)(257,606)(258,605)(259,604)(260,603)(261,602)(262,601)(263,600)(264,599)(265,598)(266,597)(267,596)(268,595)(269,594)(270,593)(271,592)(272,591)(273,590)(274,589)(275,588)(276,587)(277,586)(278,585)(279,584)(280,583)(281,582)(282,581)(283,580)(284,579)(285,578)(286,577)(287,576)(288,575)(289,574)(290,573)(291,572)(292,571)(293,570)(294,569)(295,568)(296,567)(297,566)(298,565)(299,564)(300,563)(301,562)(302,561)(303,560)(304,559)(305,558)(306,557)(307,556)(308,555)(309,554)(310,553)(311,552)(312,551)(313,550)(314,549)(315,548)(316,547)(317,546)(318,545)(319,544)(320,543)(321,542)(322,541)(323,540)(324,539)(325,538)(326,537)(327,536)(328,535)(329,534)(330,533)(331,532)(332,531)(333,530)(334,529)(335,528)(336,527)(337,526)(338,525)(339,524)(340,523)(341,522)(342,521)(343,520)(344,519)(345,518)(346,517)(347,516)(348,515)(349,514)(350,513)(351,512)(352,511)(353,510)(354,509)(355,508)(356,507)(357,506)(358,505)(359,504)(360,503)(361,502)(362,501)(363,500)(364,499)(365,498)(366,497)(367,496)(368,495)(369,494)(370,493)(371,492)(372,491)(373,490)(374,489)(375,488)(376,487)(377,486)(378,485)(379,484)(380,483)(381,482)(382,481)(383,480)(384,479)(385,478)(386,477)(387,476)(388,475)(389,474)(390,473)(391,472)(392,471)(393,470)(394,469)(395,468)(396,467)(397,466)(398,465)(399,464)(400,463)(401,462)(402,461)(403,460)(404,459)(405,458)(406,457)(407,456)(408,455)(409,454)(410,453)(411,452)(412,451)(413,450)(414,449)(415,448)(416,447)(417,446)(418,445)(419,444)(420,443)(421,442)(422,441)(423,440)(424,439)(425,438)(426,437)(427,436)(428,435)(429,434)(430,433)(431,432)
Polycyclic presentation:
2^3,3,839|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^838
Conjugacy classes & character table
5037 conjugacy classes. The character table is too large to display.
Relationships
GAP
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 839]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^838);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);
Finite Group Databasebeta