D₂₆₆₃₁ #72289976

order	53262
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₆₃₁

Splittings

Fitting, derived, Hall	C₉×C₁₁×C₂₆₉ ⋊ C₂
Hall	C₁₁×C₂₆₉ ⋊ D₉
Hall	C₉×C₂₆₉ ⋊ D₁₁
Sylow	C₂₆₉ ⋊ D₉₉
Hall	C₉×C₁₁ ⋊ D₂₆₉
Sylow	C₁₁ ⋊ D₂₄₂₁
Sylow	C₉ ⋊ D₂₉₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₁×C₂₆₉	C₂
Frattini	C₃	D₈₈₇₇
socle	C₃×C₁₁×C₂₆₉	D₃
LCS	G ▷ C₉×C₁₁×C₂₆₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 289
transitive degree	μ_t(G) = 26631

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

Polycyclic presentation:

2,3,11,269,3|b^3=e,b^a=b^2e^2,c^a=c^10,d^a=d^268,e^a=e^2

Conjugacy classes & character table

13317 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");;
gap> coll := SingleCollector(F, [2, 3, 11, 269, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^268);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26631 #72289976