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D13660 #72235552
Aliases
Splittings
Subgroups
32 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 692 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);(2,4)(6,9)(7,8)(11,692)(12,691)(13,690)(14,689)(15,688)(16,687)(17,686)(18,685)(19,684)(20,683)(21,682)(22,681)(23,680)(24,679)(25,678)(26,677)(27,676)(28,675)(29,674)(30,673)(31,672)(32,671)(33,670)(34,669)(35,668)(36,667)(37,666)(38,665)(39,664)(40,663)(41,662)(42,661)(43,660)(44,659)(45,658)(46,657)(47,656)(48,655)(49,654)(50,653)(51,652)(52,651)(53,650)(54,649)(55,648)(56,647)(57,646)(58,645)(59,644)(60,643)(61,642)(62,641)(63,640)(64,639)(65,638)(66,637)(67,636)(68,635)(69,634)(70,633)(71,632)(72,631)(73,630)(74,629)(75,628)(76,627)(77,626)(78,625)(79,624)(80,623)(81,622)(82,621)(83,620)(84,619)(85,618)(86,617)(87,616)(88,615)(89,614)(90,613)(91,612)(92,611)(93,610)(94,609)(95,608)(96,607)(97,606)(98,605)(99,604)(100,603)(101,602)(102,601)(103,600)(104,599)(105,598)(106,597)(107,596)(108,595)(109,594)(110,593)(111,592)(112,591)(113,590)(114,589)(115,588)(116,587)(117,586)(118,585)(119,584)(120,583)(121,582)(122,581)(123,580)(124,579)(125,578)(126,577)(127,576)(128,575)(129,574)(130,573)(131,572)(132,571)(133,570)(134,569)(135,568)(136,567)(137,566)(138,565)(139,564)(140,563)(141,562)(142,561)(143,560)(144,559)(145,558)(146,557)(147,556)(148,555)(149,554)(150,553)(151,552)(152,551)(153,550)(154,549)(155,548)(156,547)(157,546)(158,545)(159,544)(160,543)(161,542)(162,541)(163,540)(164,539)(165,538)(166,537)(167,536)(168,535)(169,534)(170,533)(171,532)(172,531)(173,530)(174,529)(175,528)(176,527)(177,526)(178,525)(179,524)(180,523)(181,522)(182,521)(183,520)(184,519)(185,518)(186,517)(187,516)(188,515)(189,514)(190,513)(191,512)(192,511)(193,510)(194,509)(195,508)(196,507)(197,506)(198,505)(199,504)(200,503)(201,502)(202,501)(203,500)(204,499)(205,498)(206,497)(207,496)(208,495)(209,494)(210,493)(211,492)(212,491)(213,490)(214,489)(215,488)(216,487)(217,486)(218,485)(219,484)(220,483)(221,482)(222,481)(223,480)(224,479)(225,478)(226,477)(227,476)(228,475)(229,474)(230,473)(231,472)(232,471)(233,470)(234,469)(235,468)(236,467)(237,466)(238,465)(239,464)(240,463)(241,462)(242,461)(243,460)(244,459)(245,458)(246,457)(247,456)(248,455)(249,454)(250,453)(251,452)(252,451)(253,450)(254,449)(255,448)(256,447)(257,446)(258,445)(259,444)(260,443)(261,442)(262,441)(263,440)(264,439)(265,438)(266,437)(267,436)(268,435)(269,434)(270,433)(271,432)(272,431)(273,430)(274,429)(275,428)(276,427)(277,426)(278,425)(279,424)(280,423)(281,422)(282,421)(283,420)(284,419)(285,418)(286,417)(287,416)(288,415)(289,414)(290,413)(291,412)(292,411)(293,410)(294,409)(295,408)(296,407)(297,406)(298,405)(299,404)(300,403)(301,402)(302,401)(303,400)(304,399)(305,398)(306,397)(307,396)(308,395)(309,394)(310,393)(311,392)(312,391)(313,390)(314,389)(315,388)(316,387)(317,386)(318,385)(319,384)(320,383)(321,382)(322,381)(323,380)(324,379)(325,378)(326,377)(327,376)(328,375)(329,374)(330,373)(331,372)(332,371)(333,370)(334,369)(335,368)(336,367)(337,366)(338,365)(339,364)(340,363)(341,362)(342,361)(343,360)(344,359)(345,358)(346,357)(347,356)(348,355)(349,354)(350,353)(351,352)
Polycyclic presentation:
2^3,5,683|b^2=c,b^a=bc,d^a=d^4,e^a=e^682
Conjugacy classes & character table
6833 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, 5, 683]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^682);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);
Finite Group Databasebeta