D₃₂₁₆₀ #72301942

order	64320
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₁₆₀

Splittings

Fitting	C₃×C₅×C₃₂×C₆₇ ⋊ C₂
Hall	C₃×C₅×C₆₇ ⋊ D₃₂
Hall	C₅×C₆₇ ⋊ D₉₆
Hall	C₃×C₆₇ ⋊ D₁₆₀
Sylow	C₆₇ ⋊ D₄₈₀
Hall	C₃×C₅ ⋊ D₂₁₄₄
Sylow	C₅ ⋊ D₆₄₃₂
Sylow	C₃ ⋊ D₁₀₇₂₀

Subgroups

	subgroup	G/H
center	C₂	D₁₆₀₈₀
commutator, cosocle	C₃×C₅×C₁₆×C₆₇	C₂²
Frattini	C₁₆	C₂×D₁₀₀₅
Fitting	C₃×C₅×C₃₂×C₆₇	C₂
socle	C₂×C₃×C₅×C₆₇	D₁₆
hypercenter	C₃₂	D₁₀₀₅
hypocenter	C₃×C₅×C₆₇	D₃₂
LCS	G ▷ C₃×C₅×C₁₆×C₆₇ ▷ C₃×C₅×C₈×C₆₇ ▷ C₃×C₄×C₅×C₆₇ ▷ C₂×C₃×C₅×C₆₇ ▷ C₃×C₅×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 32160

A minimal faithful permutation representation — 2 generators on 107 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,35)(37,40)(38,39)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)

Polycyclic presentation:

2^6,3,5,67|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=h^4,i^a=i^66

Conjugacy classes & character table

16083 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", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 5, 67]);;
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);;
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, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.9^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32160 #72301942