Finite Group Databasebeta

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D₃₁₃₇₅ #72301067

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

Aliases

Splittings

Subgroups

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 376 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);(2,125)(3,124)(4,123)(5,122)(6,121)(7,120)(8,119)(9,118)(10,117)(11,116)(12,115)(13,114)(14,113)(15,112)(16,111)(17,110)(18,109)(19,108)(20,107)(21,106)(22,105)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(127,376)(128,375)(129,374)(130,373)(131,372)(132,371)(133,370)(134,369)(135,368)(136,367)(137,366)(138,365)(139,364)(140,363)(141,362)(142,361)(143,360)(144,359)(145,358)(146,357)(147,356)(148,355)(149,354)(150,353)(151,352)(152,351)(153,350)(154,349)(155,348)(156,347)(157,346)(158,345)(159,344)(160,343)(161,342)(162,341)(163,340)(164,339)(165,338)(166,337)(167,336)(168,335)(169,334)(170,333)(171,332)(172,331)(173,330)(174,329)(175,328)(176,327)(177,326)(178,325)(179,324)(180,323)(181,322)(182,321)(183,320)(184,319)(185,318)(186,317)(187,316)(188,315)(189,314)(190,313)(191,312)(192,311)(193,310)(194,309)(195,308)(196,307)(197,306)(198,305)(199,304)(200,303)(201,302)(202,301)(203,300)(204,299)(205,298)(206,297)(207,296)(208,295)(209,294)(210,293)(211,292)(212,291)(213,290)(214,289)(215,288)(216,287)(217,286)(218,285)(219,284)(220,283)(221,282)(222,281)(223,280)(224,279)(225,278)(226,277)(227,276)(228,275)(229,274)(230,273)(231,272)(232,271)(233,270)(234,269)(235,268)(236,267)(237,266)(238,265)(239,264)(240,263)(241,262)(242,261)(243,260)(244,259)(245,258)(246,257)(247,256)(248,255)(249,254)(250,253)(251,252)

Polycyclic presentation:

2,5,251,5^2|b^5=d,d^5=e,b^a=b^4d^4e^4,c^a=c^250,d^a=d^4e^4,e^a=e^4

Conjugacy classes & character table

15689 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, 5, 251, 5, 5]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.4^4*F.5^4);;
gap> SetConjugate(coll, 3, 1, F.3^250);;
gap> SetConjugate(coll, 4, 1, F.4^4*F.5^4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);