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D₂₄₂₆₄ #72287166

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

Aliases

Splittings

Subgroups

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 354 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,354)(20,353)(21,352)(22,351)(23,350)(24,349)(25,348)(26,347)(27,346)(28,345)(29,344)(30,343)(31,342)(32,341)(33,340)(34,339)(35,338)(36,337)(37,336)(38,335)(39,334)(40,333)(41,332)(42,331)(43,330)(44,329)(45,328)(46,327)(47,326)(48,325)(49,324)(50,323)(51,322)(52,321)(53,320)(54,319)(55,318)(56,317)(57,316)(58,315)(59,314)(60,313)(61,312)(62,311)(63,310)(64,309)(65,308)(66,307)(67,306)(68,305)(69,304)(70,303)(71,302)(72,301)(73,300)(74,299)(75,298)(76,297)(77,296)(78,295)(79,294)(80,293)(81,292)(82,291)(83,290)(84,289)(85,288)(86,287)(87,286)(88,285)(89,284)(90,283)(91,282)(92,281)(93,280)(94,279)(95,278)(96,277)(97,276)(98,275)(99,274)(100,273)(101,272)(102,271)(103,270)(104,269)(105,268)(106,267)(107,266)(108,265)(109,264)(110,263)(111,262)(112,261)(113,260)(114,259)(115,258)(116,257)(117,256)(118,255)(119,254)(120,253)(121,252)(122,251)(123,250)(124,249)(125,248)(126,247)(127,246)(128,245)(129,244)(130,243)(131,242)(132,241)(133,240)(134,239)(135,238)(136,237)(137,236)(138,235)(139,234)(140,233)(141,232)(142,231)(143,230)(144,229)(145,228)(146,227)(147,226)(148,225)(149,224)(150,223)(151,222)(152,221)(153,220)(154,219)(155,218)(156,217)(157,216)(158,215)(159,214)(160,213)(161,212)(162,211)(163,210)(164,209)(165,208)(166,207)(167,206)(168,205)(169,204)(170,203)(171,202)(172,201)(173,200)(174,199)(175,198)(176,197)(177,196)(178,195)(179,194)(180,193)(181,192)(182,191)(183,190)(184,189)(185,188)(186,187)

Polycyclic presentation:

2^4,3,337,3|b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^336,g^a=g^2

Conjugacy classes & character table

12135 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 337, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^336);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);