Finite Group Databasebeta

?

← Prev (#72295664) Next (#72295666) →

D₃₀₃₇₁ #72295665

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

Aliases

Splittings

Subgroups

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 372 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);(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(123,372)(124,371)(125,370)(126,369)(127,368)(128,367)(129,366)(130,365)(131,364)(132,363)(133,362)(134,361)(135,360)(136,359)(137,358)(138,357)(139,356)(140,355)(141,354)(142,353)(143,352)(144,351)(145,350)(146,349)(147,348)(148,347)(149,346)(150,345)(151,344)(152,343)(153,342)(154,341)(155,340)(156,339)(157,338)(158,337)(159,336)(160,335)(161,334)(162,333)(163,332)(164,331)(165,330)(166,329)(167,328)(168,327)(169,326)(170,325)(171,324)(172,323)(173,322)(174,321)(175,320)(176,319)(177,318)(178,317)(179,316)(180,315)(181,314)(182,313)(183,312)(184,311)(185,310)(186,309)(187,308)(188,307)(189,306)(190,305)(191,304)(192,303)(193,302)(194,301)(195,300)(196,299)(197,298)(198,297)(199,296)(200,295)(201,294)(202,293)(203,292)(204,291)(205,290)(206,289)(207,288)(208,287)(209,286)(210,285)(211,284)(212,283)(213,282)(214,281)(215,280)(216,279)(217,278)(218,277)(219,276)(220,275)(221,274)(222,273)(223,272)(224,271)(225,270)(226,269)(227,268)(228,267)(229,266)(230,265)(231,264)(232,263)(233,262)(234,261)(235,260)(236,259)(237,258)(238,257)(239,256)(240,255)(241,254)(242,253)(243,252)(244,251)(245,250)(246,249)(247,248)

Polycyclic presentation:

2,11,251,11|b^11=d,b^a=b^10d^10,c^a=c^250,d^a=d^10

Conjugacy classes & character table

15187 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");;
gap> coll := SingleCollector(F, [2, 11, 251, 11]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2^10*F.4^10);;
gap> SetConjugate(coll, 3, 1, F.3^250);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);