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D₃₀₂₁₃ #72295534

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem
• Frobenius

Aliases

Splittings

Subgroups

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 454 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);(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(83,454)(84,453)(85,452)(86,451)(87,450)(88,449)(89,448)(90,447)(91,446)(92,445)(93,444)(94,443)(95,442)(96,441)(97,440)(98,439)(99,438)(100,437)(101,436)(102,435)(103,434)(104,433)(105,432)(106,431)(107,430)(108,429)(109,428)(110,427)(111,426)(112,425)(113,424)(114,423)(115,422)(116,421)(117,420)(118,419)(119,418)(120,417)(121,416)(122,415)(123,414)(124,413)(125,412)(126,411)(127,410)(128,409)(129,408)(130,407)(131,406)(132,405)(133,404)(134,403)(135,402)(136,401)(137,400)(138,399)(139,398)(140,397)(141,396)(142,395)(143,394)(144,393)(145,392)(146,391)(147,390)(148,389)(149,388)(150,387)(151,386)(152,385)(153,384)(154,383)(155,382)(156,381)(157,380)(158,379)(159,378)(160,377)(161,376)(162,375)(163,374)(164,373)(165,372)(166,371)(167,370)(168,369)(169,368)(170,367)(171,366)(172,365)(173,364)(174,363)(175,362)(176,361)(177,360)(178,359)(179,358)(180,357)(181,356)(182,355)(183,354)(184,353)(185,352)(186,351)(187,350)(188,349)(189,348)(190,347)(191,346)(192,345)(193,344)(194,343)(195,342)(196,341)(197,340)(198,339)(199,338)(200,337)(201,336)(202,335)(203,334)(204,333)(205,332)(206,331)(207,330)(208,329)(209,328)(210,327)(211,326)(212,325)(213,324)(214,323)(215,322)(216,321)(217,320)(218,319)(219,318)(220,317)(221,316)(222,315)(223,314)(224,313)(225,312)(226,311)(227,310)(228,309)(229,308)(230,307)(231,306)(232,305)(233,304)(234,303)(235,302)(236,301)(237,300)(238,299)(239,298)(240,297)(241,296)(242,295)(243,294)(244,293)(245,292)(246,291)(247,290)(248,289)(249,288)(250,287)(251,286)(252,285)(253,284)(254,283)(255,282)(256,281)(257,280)(258,279)(259,278)(260,277)(261,276)(262,275)(263,274)(264,273)(265,272)(266,271)(267,270)(268,269)

Polycyclic presentation:

2,3,373,3^3|b^3=d,d^3=e,e^3=f,b^a=b^2d^2e^2f^2,c^a=c^372,d^a=d^2e^2f^2,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

15108 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, 3, 373, 3, 3, 3]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.4^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^372);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.5^2*F.6^2);;
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);