Finite Group Databasebeta

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D₂₂₇₆₁ #72277726

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

Polycyclic presentation:

2,3,281,3^3|b^3=d,d^3=e,e^3=f,b^a=b^2d^2e^2f^2,c^a=c^280,d^a=d^2e^2f^2,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

11382 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, 281, 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.3^280);;
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);