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Dic₁₁₀₄₃ #72275153

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial

Aliases

Splittings

Subgroups

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 440 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);(1,3,2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,440)(34,439)(35,438)(36,437)(37,436)(38,435)(39,434)(40,433)(41,432)(42,431)(43,430)(44,429)(45,428)(46,427)(47,426)(48,425)(49,424)(50,423)(51,422)(52,421)(53,420)(54,419)(55,418)(56,417)(57,416)(58,415)(59,414)(60,413)(61,412)(62,411)(63,410)(64,409)(65,408)(66,407)(67,406)(68,405)(69,404)(70,403)(71,402)(72,401)(73,400)(74,399)(75,398)(76,397)(77,396)(78,395)(79,394)(80,393)(81,392)(82,391)(83,390)(84,389)(85,388)(86,387)(87,386)(88,385)(89,384)(90,383)(91,382)(92,381)(93,380)(94,379)(95,378)(96,377)(97,376)(98,375)(99,374)(100,373)(101,372)(102,371)(103,370)(104,369)(105,368)(106,367)(107,366)(108,365)(109,364)(110,363)(111,362)(112,361)(113,360)(114,359)(115,358)(116,357)(117,356)(118,355)(119,354)(120,353)(121,352)(122,351)(123,350)(124,349)(125,348)(126,347)(127,346)(128,345)(129,344)(130,343)(131,342)(132,341)(133,340)(134,339)(135,338)(136,337)(137,336)(138,335)(139,334)(140,333)(141,332)(142,331)(143,330)(144,329)(145,328)(146,327)(147,326)(148,325)(149,324)(150,323)(151,322)(152,321)(153,320)(154,319)(155,318)(156,317)(157,316)(158,315)(159,314)(160,313)(161,312)(162,311)(163,310)(164,309)(165,308)(166,307)(167,306)(168,305)(169,304)(170,303)(171,302)(172,301)(173,300)(174,299)(175,298)(176,297)(177,296)(178,295)(179,294)(180,293)(181,292)(182,291)(183,290)(184,289)(185,288)(186,287)(187,286)(188,285)(189,284)(190,283)(191,282)(192,281)(193,280)(194,279)(195,278)(196,277)(197,276)(198,275)(199,274)(200,273)(201,272)(202,271)(203,270)(204,269)(205,268)(206,267)(207,266)(208,265)(209,264)(210,263)(211,262)(212,261)(213,260)(214,259)(215,258)(216,257)(217,256)(218,255)(219,254)(220,253)(221,252)(222,251)(223,250)(224,249)(225,248)(226,247)(227,246)(228,245)(229,244)(230,243)(231,242)(232,241)(233,240)(234,239)(235,238)(236,237)

Polycyclic presentation:

2^2,3,409,3^2|a^2=b,c^3=e,e^3=f,c^a=c^2e^2f^2,d^a=d^408,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

11046 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", "f");;
gap> coll := SingleCollector(F, [2, 2, 3, 409, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4^408);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);