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D₃₀₁₂₅ #72295439

• 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 366 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);(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)(127,366)(128,365)(129,364)(130,363)(131,362)(132,361)(133,360)(134,359)(135,358)(136,357)(137,356)(138,355)(139,354)(140,353)(141,352)(142,351)(143,350)(144,349)(145,348)(146,347)(147,346)(148,345)(149,344)(150,343)(151,342)(152,341)(153,340)(154,339)(155,338)(156,337)(157,336)(158,335)(159,334)(160,333)(161,332)(162,331)(163,330)(164,329)(165,328)(166,327)(167,326)(168,325)(169,324)(170,323)(171,322)(172,321)(173,320)(174,319)(175,318)(176,317)(177,316)(178,315)(179,314)(180,313)(181,312)(182,311)(183,310)(184,309)(185,308)(186,307)(187,306)(188,305)(189,304)(190,303)(191,302)(192,301)(193,300)(194,299)(195,298)(196,297)(197,296)(198,295)(199,294)(200,293)(201,292)(202,291)(203,290)(204,289)(205,288)(206,287)(207,286)(208,285)(209,284)(210,283)(211,282)(212,281)(213,280)(214,279)(215,278)(216,277)(217,276)(218,275)(219,274)(220,273)(221,272)(222,271)(223,270)(224,269)(225,268)(226,267)(227,266)(228,265)(229,264)(230,263)(231,262)(232,261)(233,260)(234,259)(235,258)(236,257)(237,256)(238,255)(239,254)(240,253)(241,252)(242,251)(243,250)(244,249)(245,248)(246,247)

Polycyclic presentation:

2,5,241,5^2|b^5=d,d^5=e,b^a=b^4d^4e^4,c^a=c^240,d^a=d^4e^4,e^a=e^4

Conjugacy classes & character table

15064 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, 241, 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.3^240);;
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);