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Dic4854 #72216360
Aliases
Splittings
Subgroups
24 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 820 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);(1,5,3,7)(2,6,4,8)(10,11)(13,820)(14,819)(15,818)(16,817)(17,816)(18,815)(19,814)(20,813)(21,812)(22,811)(23,810)(24,809)(25,808)(26,807)(27,806)(28,805)(29,804)(30,803)(31,802)(32,801)(33,800)(34,799)(35,798)(36,797)(37,796)(38,795)(39,794)(40,793)(41,792)(42,791)(43,790)(44,789)(45,788)(46,787)(47,786)(48,785)(49,784)(50,783)(51,782)(52,781)(53,780)(54,779)(55,778)(56,777)(57,776)(58,775)(59,774)(60,773)(61,772)(62,771)(63,770)(64,769)(65,768)(66,767)(67,766)(68,765)(69,764)(70,763)(71,762)(72,761)(73,760)(74,759)(75,758)(76,757)(77,756)(78,755)(79,754)(80,753)(81,752)(82,751)(83,750)(84,749)(85,748)(86,747)(87,746)(88,745)(89,744)(90,743)(91,742)(92,741)(93,740)(94,739)(95,738)(96,737)(97,736)(98,735)(99,734)(100,733)(101,732)(102,731)(103,730)(104,729)(105,728)(106,727)(107,726)(108,725)(109,724)(110,723)(111,722)(112,721)(113,720)(114,719)(115,718)(116,717)(117,716)(118,715)(119,714)(120,713)(121,712)(122,711)(123,710)(124,709)(125,708)(126,707)(127,706)(128,705)(129,704)(130,703)(131,702)(132,701)(133,700)(134,699)(135,698)(136,697)(137,696)(138,695)(139,694)(140,693)(141,692)(142,691)(143,690)(144,689)(145,688)(146,687)(147,686)(148,685)(149,684)(150,683)(151,682)(152,681)(153,680)(154,679)(155,678)(156,677)(157,676)(158,675)(159,674)(160,673)(161,672)(162,671)(163,670)(164,669)(165,668)(166,667)(167,666)(168,665)(169,664)(170,663)(171,662)(172,661)(173,660)(174,659)(175,658)(176,657)(177,656)(178,655)(179,654)(180,653)(181,652)(182,651)(183,650)(184,649)(185,648)(186,647)(187,646)(188,645)(189,644)(190,643)(191,642)(192,641)(193,640)(194,639)(195,638)(196,637)(197,636)(198,635)(199,634)(200,633)(201,632)(202,631)(203,630)(204,629)(205,628)(206,627)(207,626)(208,625)(209,624)(210,623)(211,622)(212,621)(213,620)(214,619)(215,618)(216,617)(217,616)(218,615)(219,614)(220,613)(221,612)(222,611)(223,610)(224,609)(225,608)(226,607)(227,606)(228,605)(229,604)(230,603)(231,602)(232,601)(233,600)(234,599)(235,598)(236,597)(237,596)(238,595)(239,594)(240,593)(241,592)(242,591)(243,590)(244,589)(245,588)(246,587)(247,586)(248,585)(249,584)(250,583)(251,582)(252,581)(253,580)(254,579)(255,578)(256,577)(257,576)(258,575)(259,574)(260,573)(261,572)(262,571)(263,570)(264,569)(265,568)(266,567)(267,566)(268,565)(269,564)(270,563)(271,562)(272,561)(273,560)(274,559)(275,558)(276,557)(277,556)(278,555)(279,554)(280,553)(281,552)(282,551)(283,550)(284,549)(285,548)(286,547)(287,546)(288,545)(289,544)(290,543)(291,542)(292,541)(293,540)(294,539)(295,538)(296,537)(297,536)(298,535)(299,534)(300,533)(301,532)(302,531)(303,530)(304,529)(305,528)(306,527)(307,526)(308,525)(309,524)(310,523)(311,522)(312,521)(313,520)(314,519)(315,518)(316,517)(317,516)(318,515)(319,514)(320,513)(321,512)(322,511)(323,510)(324,509)(325,508)(326,507)(327,506)(328,505)(329,504)(330,503)(331,502)(332,501)(333,500)(334,499)(335,498)(336,497)(337,496)(338,495)(339,494)(340,493)(341,492)(342,491)(343,490)(344,489)(345,488)(346,487)(347,486)(348,485)(349,484)(350,483)(351,482)(352,481)(353,480)(354,479)(355,478)(356,477)(357,476)(358,475)(359,474)(360,473)(361,472)(362,471)(363,470)(364,469)(365,468)(366,467)(367,466)(368,465)(369,464)(370,463)(371,462)(372,461)(373,460)(374,459)(375,458)(376,457)(377,456)(378,455)(379,454)(380,453)(381,452)(382,451)(383,450)(384,449)(385,448)(386,447)(387,446)(388,445)(389,444)(390,443)(391,442)(392,441)(393,440)(394,439)(395,438)(396,437)(397,436)(398,435)(399,434)(400,433)(401,432)(402,431)(403,430)(404,429)(405,428)(406,427)(407,426)(408,425)(409,424)(410,423)(411,422)(412,421)(413,420)(414,419)(415,418)(416,417)
Polycyclic presentation:
2^3,3,809|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^808
Conjugacy classes & character table
4857 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, 809]);;
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.5^808);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);
Finite Group Databasebeta