D₁₁₁₃₆ #72215852

order	22272
automorphism	Aut(G) order 39911424
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₃₆

Splittings

Fitting	C₃×C₂₉×C₁₂₈ ⋊ C₂
Hall	C₃×C₂₉ ⋊ D₁₂₈
Sylow	C₂₉ ⋊ D₃₈₄
Sylow	C₃ ⋊ D₃₇₁₂

Subgroups

	subgroup	G/H
center	C₂	D₅₅₆₈
commutator, cosocle	C₃×C₂₉×C₆₄	C₂²
Frattini	C₆₄	C₂×D₈₇
Fitting	C₃×C₂₉×C₁₂₈	C₂
socle	C₂×C₃×C₂₉	D₆₄
hypercenter	C₁₂₈	D₈₇
hypocenter	C₃×C₂₉	D₁₂₈
LCS	G ▷ C₃×C₂₉×C₆₄ ▷ C₃×C₂₉×C₃₂ ▷ C₃×C₁₆×C₂₉ ▷ C₃×C₈×C₂₉ ▷ C₃×C₄×C₂₉ ▷ C₂×C₃×C₂₉ ▷ C₃×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄ ◁ C₁₂₈

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 160
transitive degree	μ_t(G) = 11136

A minimal faithful permutation representation — 2 generators on 160 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);(2,128)(3,127)(4,126)(5,125)(6,124)(7,123)(8,122)(9,121)(10,120)(11,119)(12,118)(13,117)(14,116)(15,115)(16,114)(17,113)(18,112)(19,111)(20,110)(21,109)(22,108)(23,107)(24,106)(25,105)(26,104)(27,103)(28,102)(29,101)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,70)(61,69)(62,68)(63,67)(64,66)(130,131)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)

Polycyclic presentation:

2^8,3,29|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,i^a=i^2,j^a=j^28

Conjugacy classes & character table

5571 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₂₇₂
maximal normal subgroup of	D₂₂₂₇₂
maximal quotient of	Dic₁₁₁₃₆, D₂₂₂₇₂

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 3, 29]);;
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);;
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, 9, 1, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11136 #72215852