D₂₆₈₂₉ #72290168

order	53658
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₆₈₂₉

Splittings

Fitting, derived, Hall	C₉×C₁₁×C₂₇₁ ⋊ C₂
Hall	C₁₁×C₂₇₁ ⋊ D₉
Hall	C₉×C₂₇₁ ⋊ D₁₁
Sylow	C₂₇₁ ⋊ D₉₉
Hall	C₉×C₁₁ ⋊ D₂₇₁
Sylow	C₁₁ ⋊ D₂₄₃₉
Sylow	C₉ ⋊ D₂₉₈₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₁×C₂₇₁	C₂
Frattini	C₃	D₈₉₄₃
socle	C₃×C₁₁×C₂₇₁	D₃
LCS	G ▷ C₉×C₁₁×C₂₇₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 291
transitive degree	μ_t(G) = 26829

A minimal faithful permutation representation — 2 generators on 291 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);(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(22,291)(23,290)(24,289)(25,288)(26,287)(27,286)(28,285)(29,284)(30,283)(31,282)(32,281)(33,280)(34,279)(35,278)(36,277)(37,276)(38,275)(39,274)(40,273)(41,272)(42,271)(43,270)(44,269)(45,268)(46,267)(47,266)(48,265)(49,264)(50,263)(51,262)(52,261)(53,260)(54,259)(55,258)(56,257)(57,256)(58,255)(59,254)(60,253)(61,252)(62,251)(63,250)(64,249)(65,248)(66,247)(67,246)(68,245)(69,244)(70,243)(71,242)(72,241)(73,240)(74,239)(75,238)(76,237)(77,236)(78,235)(79,234)(80,233)(81,232)(82,231)(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:

2,3,11,271,3|b^3=e,b^a=b^2e^2,c^a=c^10,d^a=d^270,e^a=e^2

Conjugacy classes & character table

13416 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, 271, 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.4^270);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26829 #72290168