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D₁₂₀₆₉ #72224028

• 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 230 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);(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)(83,230)(84,229)(85,228)(86,227)(87,226)(88,225)(89,224)(90,223)(91,222)(92,221)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)(112,201)(113,200)(114,199)(115,198)(116,197)(117,196)(118,195)(119,194)(120,193)(121,192)(122,191)(123,190)(124,189)(125,188)(126,187)(127,186)(128,185)(129,184)(130,183)(131,182)(132,181)(133,180)(134,179)(135,178)(136,177)(137,176)(138,175)(139,174)(140,173)(141,172)(142,171)(143,170)(144,169)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,161)(153,160)(154,159)(155,158)(156,157)

Polycyclic presentation:

2,3,149,3^3|b^3=d,d^3=e,e^3=f,b^a=b^2d^2e^2f^2,c^a=c^148,d^a=d^2e^2f^2,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

6036 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");;
gap> coll := SingleCollector(F, [2, 3, 149, 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.3^148);;
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);