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Dic₁₁₅₀₆ #72276604

• 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 542 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);(1,5,3,7)(2,6,4,8)(10,19)(11,18)(12,17)(13,16)(14,15)(21,542)(22,541)(23,540)(24,539)(25,538)(26,537)(27,536)(28,535)(29,534)(30,533)(31,532)(32,531)(33,530)(34,529)(35,528)(36,527)(37,526)(38,525)(39,524)(40,523)(41,522)(42,521)(43,520)(44,519)(45,518)(46,517)(47,516)(48,515)(49,514)(50,513)(51,512)(52,511)(53,510)(54,509)(55,508)(56,507)(57,506)(58,505)(59,504)(60,503)(61,502)(62,501)(63,500)(64,499)(65,498)(66,497)(67,496)(68,495)(69,494)(70,493)(71,492)(72,491)(73,490)(74,489)(75,488)(76,487)(77,486)(78,485)(79,484)(80,483)(81,482)(82,481)(83,480)(84,479)(85,478)(86,477)(87,476)(88,475)(89,474)(90,473)(91,472)(92,471)(93,470)(94,469)(95,468)(96,467)(97,466)(98,465)(99,464)(100,463)(101,462)(102,461)(103,460)(104,459)(105,458)(106,457)(107,456)(108,455)(109,454)(110,453)(111,452)(112,451)(113,450)(114,449)(115,448)(116,447)(117,446)(118,445)(119,444)(120,443)(121,442)(122,441)(123,440)(124,439)(125,438)(126,437)(127,436)(128,435)(129,434)(130,433)(131,432)(132,431)(133,430)(134,429)(135,428)(136,427)(137,426)(138,425)(139,424)(140,423)(141,422)(142,421)(143,420)(144,419)(145,418)(146,417)(147,416)(148,415)(149,414)(150,413)(151,412)(152,411)(153,410)(154,409)(155,408)(156,407)(157,406)(158,405)(159,404)(160,403)(161,402)(162,401)(163,400)(164,399)(165,398)(166,397)(167,396)(168,395)(169,394)(170,393)(171,392)(172,391)(173,390)(174,389)(175,388)(176,387)(177,386)(178,385)(179,384)(180,383)(181,382)(182,381)(183,380)(184,379)(185,378)(186,377)(187,376)(188,375)(189,374)(190,373)(191,372)(192,371)(193,370)(194,369)(195,368)(196,367)(197,366)(198,365)(199,364)(200,363)(201,362)(202,361)(203,360)(204,359)(205,358)(206,357)(207,356)(208,355)(209,354)(210,353)(211,352)(212,351)(213,350)(214,349)(215,348)(216,347)(217,346)(218,345)(219,344)(220,343)(221,342)(222,341)(223,340)(224,339)(225,338)(226,337)(227,336)(228,335)(229,334)(230,333)(231,332)(232,331)(233,330)(234,329)(235,328)(236,327)(237,326)(238,325)(239,324)(240,323)(241,322)(242,321)(243,320)(244,319)(245,318)(246,317)(247,316)(248,315)(249,314)(250,313)(251,312)(252,311)(253,310)(254,309)(255,308)(256,307)(257,306)(258,305)(259,304)(260,303)(261,302)(262,301)(263,300)(264,299)(265,298)(266,297)(267,296)(268,295)(269,294)(270,293)(271,292)(272,291)(273,290)(274,289)(275,288)(276,287)(277,286)(278,285)(279,284)(280,283)(281,282)

Polycyclic presentation:

2^3,11,523|a^2=c,b^2=c,b^a=bc,d^a=d^10,e^a=e^522

Conjugacy classes & character table

11509 conjugacy classes. The character table is too large to display.

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, 11, 523]);;
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.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^522);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);