D₈₁₆₀ #72213188

order	16320
automorphism	Aut(G) order 16711680
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₁₆₀

Splittings

Fitting	C₃×C₅×C₁₇×C₃₂ ⋊ C₂
Hall	C₃×C₅×C₁₇ ⋊ D₃₂
Hall	C₅×C₁₇ ⋊ D₉₆
Hall	C₃×C₁₇ ⋊ D₁₆₀
Sylow	C₁₇ ⋊ D₄₈₀
Hall	C₃×C₅ ⋊ D₅₄₄
Sylow	C₅ ⋊ D₁₆₃₂
Sylow	C₃ ⋊ D₂₇₂₀

Subgroups

	subgroup	G/H
center	C₂	D₄₀₈₀
commutator, cosocle	C₃×C₅×C₁₆×C₁₇	C₂²
Frattini	C₁₆	C₂×D₂₅₅
Fitting	C₃×C₅×C₁₇×C₃₂	C₂
socle	C₂×C₃×C₅×C₁₇	D₁₆
hypercenter	C₃₂	D₂₅₅
hypocenter	C₃×C₅×C₁₇	D₃₂
LCS	G ▷ C₃×C₅×C₁₆×C₁₇ ▷ C₃×C₅×C₈×C₁₇ ▷ C₃×C₄×C₅×C₁₇ ▷ C₂×C₃×C₅×C₁₇ ▷ C₃×C₅×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 8160

A minimal faithful permutation representation — 2 generators on 57 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);(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)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2^6,3,5,17|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^16

Conjugacy classes & character table

4083 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₆₃₂₀, D₂₄₄₈₀
maximal normal subgroup of	D₁₆₃₂₀
maximal quotient of	Dic₈₁₆₀, D₁₆₃₂₀, D₂₄₄₈₀

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, 17]);;
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.9^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8160 #72213188