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D₂₇₂₉₇ #72290616

• 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 418 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);(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)(83,418)(84,417)(85,416)(86,415)(87,414)(88,413)(89,412)(90,411)(91,410)(92,409)(93,408)(94,407)(95,406)(96,405)(97,404)(98,403)(99,402)(100,401)(101,400)(102,399)(103,398)(104,397)(105,396)(106,395)(107,394)(108,393)(109,392)(110,391)(111,390)(112,389)(113,388)(114,387)(115,386)(116,385)(117,384)(118,383)(119,382)(120,381)(121,380)(122,379)(123,378)(124,377)(125,376)(126,375)(127,374)(128,373)(129,372)(130,371)(131,370)(132,369)(133,368)(134,367)(135,366)(136,365)(137,364)(138,363)(139,362)(140,361)(141,360)(142,359)(143,358)(144,357)(145,356)(146,355)(147,354)(148,353)(149,352)(150,351)(151,350)(152,349)(153,348)(154,347)(155,346)(156,345)(157,344)(158,343)(159,342)(160,341)(161,340)(162,339)(163,338)(164,337)(165,336)(166,335)(167,334)(168,333)(169,332)(170,331)(171,330)(172,329)(173,328)(174,327)(175,326)(176,325)(177,324)(178,323)(179,322)(180,321)(181,320)(182,319)(183,318)(184,317)(185,316)(186,315)(187,314)(188,313)(189,312)(190,311)(191,310)(192,309)(193,308)(194,307)(195,306)(196,305)(197,304)(198,303)(199,302)(200,301)(201,300)(202,299)(203,298)(204,297)(205,296)(206,295)(207,294)(208,293)(209,292)(210,291)(211,290)(212,289)(213,288)(214,287)(215,286)(216,285)(217,284)(218,283)(219,282)(220,281)(221,280)(222,279)(223,278)(224,277)(225,276)(226,275)(227,274)(228,273)(229,272)(230,271)(231,270)(232,269)(233,268)(234,267)(235,266)(236,265)(237,264)(238,263)(239,262)(240,261)(241,260)(242,259)(243,258)(244,257)(245,256)(246,255)(247,254)(248,253)(249,252)(250,251)

Polycyclic presentation:

2,3,337,3^3|b^3=d,d^3=e,e^3=f,b^a=b^2d^2e^2f^2,c^a=c^336,d^a=d^2e^2f^2,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

13650 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, 337, 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.3^336);;
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);