Finite Group Databasebeta

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D₃₄₃ #10621852

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

Aliases

Splittings

Subgroups

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 343 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);(2,343)(3,342)(4,341)(5,340)(6,339)(7,338)(8,337)(9,336)(10,335)(11,334)(12,333)(13,332)(14,331)(15,330)(16,329)(17,328)(18,327)(19,326)(20,325)(21,324)(22,323)(23,322)(24,321)(25,320)(26,319)(27,318)(28,317)(29,316)(30,315)(31,314)(32,313)(33,312)(34,311)(35,310)(36,309)(37,308)(38,307)(39,306)(40,305)(41,304)(42,303)(43,302)(44,301)(45,300)(46,299)(47,298)(48,297)(49,296)(50,295)(51,294)(52,293)(53,292)(54,291)(55,290)(56,289)(57,288)(58,287)(59,286)(60,285)(61,284)(62,283)(63,282)(64,281)(65,280)(66,279)(67,278)(68,277)(69,276)(70,275)(71,274)(72,273)(73,272)(74,271)(75,270)(76,269)(77,268)(78,267)(79,266)(80,265)(81,264)(82,263)(83,262)(84,261)(85,260)(86,259)(87,258)(88,257)(89,256)(90,255)(91,254)(92,253)(93,252)(94,251)(95,250)(96,249)(97,248)(98,247)(99,246)(100,245)(101,244)(102,243)(103,242)(104,241)(105,240)(106,239)(107,238)(108,237)(109,236)(110,235)(111,234)(112,233)(113,232)(114,231)(115,230)(116,229)(117,228)(118,227)(119,226)(120,225)(121,224)(122,223)(123,222)(124,221)(125,220)(126,219)(127,218)(128,217)(129,216)(130,215)(131,214)(132,213)(133,212)(134,211)(135,210)(136,209)(137,208)(138,207)(139,206)(140,205)(141,204)(142,203)(143,202)(144,201)(145,200)(146,199)(147,198)(148,197)(149,196)(150,195)(151,194)(152,193)(153,192)(154,191)(155,190)(156,189)(157,188)(158,187)(159,186)(160,185)(161,184)(162,183)(163,182)(164,181)(165,180)(166,179)(167,178)(168,177)(169,176)(170,175)(171,174)(172,173)

Polycyclic presentation:

2,7^3|b^7=c,c^7=d,b^a=b^6c^6d^6,c^a=c^6d^6,d^a=d^6

Conjugacy classes & character table

173 conjugacy classes — view the full character table.

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(686, 1);