D₁₁₇₁₂ #72223947

order	23424
automorphism	Aut(G) order 44974080
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₇₁₂

Splittings

Fitting	C₃×C₆₁×C₆₄ ⋊ C₂
Hall	C₃×C₆₁ ⋊ D₆₄
Sylow	C₆₁ ⋊ D₁₉₂
Sylow	C₃ ⋊ D₃₉₀₄

Subgroups

	subgroup	G/H
center	C₂	D₅₈₅₆
commutator, cosocle	C₃×C₃₂×C₆₁	C₂²
Frattini	C₃₂	C₂×D₁₈₃
Fitting	C₃×C₆₁×C₆₄	C₂
socle	C₂×C₃×C₆₁	D₃₂
hypercenter	C₆₄	D₁₈₃
hypocenter	C₃×C₆₁	D₆₄
LCS	G ▷ C₃×C₃₂×C₆₁ ▷ C₃×C₁₆×C₆₁ ▷ C₃×C₈×C₆₁ ▷ C₃×C₄×C₆₁ ▷ C₂×C₃×C₆₁ ▷ C₃×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 11712

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

Polycyclic presentation:

2^7,3,61|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2,i^a=i^60

Conjugacy classes & character table

5859 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₄₂₄
maximal normal subgroup of	D₂₃₄₂₄
maximal quotient of	Dic₁₁₇₁₂, D₂₃₄₂₄

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 61]);;
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);;
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, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11712 #72223947