D₄₀₈₀ #72211291

order	8160
automorphism	Aut(G) order 4177920
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₀₈₀

Splittings

Fitting	C₃×C₅×C₁₆×C₁₇ ⋊ C₂
Hall	C₃×C₅×C₁₇ ⋊ D₁₆
Hall	C₅×C₁₇ ⋊ D₄₈
Hall	C₃×C₁₇ ⋊ D₈₀
Sylow	C₁₇ ⋊ D₂₄₀
Hall	C₃×C₅ ⋊ D₂₇₂
Sylow	C₅ ⋊ D₈₁₆
Sylow	C₃ ⋊ D₁₃₆₀

Subgroups

	subgroup	G/H
center	C₂	D₂₀₄₀
commutator, cosocle	C₃×C₅×C₈×C₁₇	C₂²
Frattini	C₈	C₂×D₂₅₅
Fitting	C₃×C₅×C₁₆×C₁₇	C₂
socle	C₂×C₃×C₅×C₁₇	D₈
hypercenter	C₁₆	D₂₅₅
hypocenter	C₃×C₅×C₁₇	D₁₆
LCS	G ▷ C₃×C₅×C₈×C₁₇ ▷ C₃×C₄×C₅×C₁₇ ▷ C₂×C₃×C₅×C₁₇ ▷ C₃×C₅×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 4080

A minimal faithful permutation representation — 2 generators on 41 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,24)(22,23)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)

Polycyclic presentation:

2^5,3,5,17|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^4,h^a=h^16

Conjugacy classes & character table

2043 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₈₁₆₀, D₁₂₂₄₀, D₂₀₄₀₀, …
maximal normal subgroup of	D₈₁₆₀
maximal quotient of	D₈₁₆₀, Dic₄₀₈₀, D₁₂₂₄₀, …

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");;
gap> coll := SingleCollector(F, [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);;
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, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4080 #72211291