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D₁₆₁₂₈ #72272511

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

Aliases

Splittings

Subgroups

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 272 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);(2,256)(3,255)(4,254)(5,253)(6,252)(7,251)(8,250)(9,249)(10,248)(11,247)(12,246)(13,245)(14,244)(15,243)(16,242)(17,241)(18,240)(19,239)(20,238)(21,237)(22,236)(23,235)(24,234)(25,233)(26,232)(27,231)(28,230)(29,229)(30,228)(31,227)(32,226)(33,225)(34,224)(35,223)(36,222)(37,221)(38,220)(39,219)(40,218)(41,217)(42,216)(43,215)(44,214)(45,213)(46,212)(47,211)(48,210)(49,209)(50,208)(51,207)(52,206)(53,205)(54,204)(55,203)(56,202)(57,201)(58,200)(59,199)(60,198)(61,197)(62,196)(63,195)(64,194)(65,193)(66,192)(67,191)(68,190)(69,189)(70,188)(71,187)(72,186)(73,185)(74,184)(75,183)(76,182)(77,181)(78,180)(79,179)(80,178)(81,177)(82,176)(83,175)(84,174)(85,173)(86,172)(87,171)(88,170)(89,169)(90,168)(91,167)(92,166)(93,165)(94,164)(95,163)(96,162)(97,161)(98,160)(99,159)(100,158)(101,157)(102,156)(103,155)(104,154)(105,153)(106,152)(107,151)(108,150)(109,149)(110,148)(111,147)(112,146)(113,145)(114,144)(115,143)(116,142)(117,141)(118,140)(119,139)(120,138)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(258,265)(259,264)(260,263)(261,262)(267,272)(268,271)(269,270)

Polycyclic presentation:

2^9,3,7,3|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=hi,h^2=i,j^3=l,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,h^a=hi,j^a=j^2l^2,k^a=k^6,l^a=l^2

Conjugacy classes & character table

8067 conjugacy classes. The character table is too large to display.

Relationships

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", "h", "i", "j", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 7, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 10, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2*F.12^2);;
gap> SetConjugate(coll, 11, 1, F.11^6);;
gap> SetConjugate(coll, 12, 1, F.12^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);