Finite Group Databasebeta

?

← Prev (#72274994) Next (#72274996) →

D₂₁₃₀₃ #72274995

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

Aliases

Splittings

Subgroups

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 344 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);(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(83,344)(84,343)(85,342)(86,341)(87,340)(88,339)(89,338)(90,337)(91,336)(92,335)(93,334)(94,333)(95,332)(96,331)(97,330)(98,329)(99,328)(100,327)(101,326)(102,325)(103,324)(104,323)(105,322)(106,321)(107,320)(108,319)(109,318)(110,317)(111,316)(112,315)(113,314)(114,313)(115,312)(116,311)(117,310)(118,309)(119,308)(120,307)(121,306)(122,305)(123,304)(124,303)(125,302)(126,301)(127,300)(128,299)(129,298)(130,297)(131,296)(132,295)(133,294)(134,293)(135,292)(136,291)(137,290)(138,289)(139,288)(140,287)(141,286)(142,285)(143,284)(144,283)(145,282)(146,281)(147,280)(148,279)(149,278)(150,277)(151,276)(152,275)(153,274)(154,273)(155,272)(156,271)(157,270)(158,269)(159,268)(160,267)(161,266)(162,265)(163,264)(164,263)(165,262)(166,261)(167,260)(168,259)(169,258)(170,257)(171,256)(172,255)(173,254)(174,253)(175,252)(176,251)(177,250)(178,249)(179,248)(180,247)(181,246)(182,245)(183,244)(184,243)(185,242)(186,241)(187,240)(188,239)(189,238)(190,237)(191,236)(192,235)(193,234)(194,233)(195,232)(196,231)(197,230)(198,229)(199,228)(200,227)(201,226)(202,225)(203,224)(204,223)(205,222)(206,221)(207,220)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214)

Polycyclic presentation:

2,3,263,3^3|b^3=d,d^3=e,e^3=f,b^a=b^2d^2e^2f^2,c^a=c^262,d^a=d^2e^2f^2,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

10653 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");;
gap> coll := SingleCollector(F, [2, 3, 263, 3, 3, 3]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.4^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^262);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);