D₁₁₈₃₆ #72223974

order	23672
automorphism	Aut(G) order 63440960
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₈₃₆

Splittings

Fitting	C₄×C₁₁×C₂₆₉ ⋊ C₂
Hall	C₁₁×C₂₆₉ ⋊ D₄
Sylow	C₂₆₉ ⋊ D₄₄
Sylow	C₁₁ ⋊ D₁₀₇₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₅₉
commutator, socle, cosocle	C₂×C₁₁×C₂₆₉	C₂²
Fitting	C₄×C₁₁×C₂₆₉	C₂
hypercenter	C₄	D₂₉₅₉
hypocenter	C₁₁×C₂₆₉	D₄
LCS	G ▷ C₂×C₁₁×C₂₆₉ ▷ C₁₁×C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 284
transitive degree	μ_t(G) = 11836

A minimal faithful permutation representation — 2 generators on 284 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);(2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,284)(18,283)(19,282)(20,281)(21,280)(22,279)(23,278)(24,277)(25,276)(26,275)(27,274)(28,273)(29,272)(30,271)(31,270)(32,269)(33,268)(34,267)(35,266)(36,265)(37,264)(38,263)(39,262)(40,261)(41,260)(42,259)(43,258)(44,257)(45,256)(46,255)(47,254)(48,253)(49,252)(50,251)(51,250)(52,249)(53,248)(54,247)(55,246)(56,245)(57,244)(58,243)(59,242)(60,241)(61,240)(62,239)(63,238)(64,237)(65,236)(66,235)(67,234)(68,233)(69,232)(70,231)(71,230)(72,229)(73,228)(74,227)(75,226)(76,225)(77,224)(78,223)(79,222)(80,221)(81,220)(82,219)(83,218)(84,217)(85,216)(86,215)(87,214)(88,213)(89,212)(90,211)(91,210)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,203)(99,202)(100,201)(101,200)(102,199)(103,198)(104,197)(105,196)(106,195)(107,194)(108,193)(109,192)(110,191)(111,190)(112,189)(113,188)(114,187)(115,186)(116,185)(117,184)(118,183)(119,182)(120,181)(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)(131,170)(132,169)(133,168)(134,167)(135,166)(136,165)(137,164)(138,163)(139,162)(140,161)(141,160)(142,159)(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)

Polycyclic presentation:

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

Conjugacy classes & character table

5921 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₆₇₂
maximal normal subgroup of	D₂₃₆₇₂
maximal quotient of	Dic₁₁₈₃₆, D₂₃₆₇₂

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

D11836 #72223974