D₂₀₄₀ #72211044

order	4080
automorphism	Aut(G) order 1044480
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₀₄₀

Splittings

Fitting	C₃×C₅×C₈×C₁₇ ⋊ C₂
Hall	C₃×C₅×C₁₇ ⋊ D₈
Hall	C₅×C₁₇ ⋊ D₂₄
Hall	C₃×C₁₇ ⋊ D₄₀
Sylow	C₁₇ ⋊ D₁₂₀
Hall	C₃×C₅ ⋊ D₁₃₆
Sylow	C₅ ⋊ D₄₀₈
Sylow	C₃ ⋊ D₆₈₀

Subgroups

	subgroup	G/H
center	C₂	D₁₀₂₀
commutator, cosocle	C₃×C₄×C₅×C₁₇	C₂²
Frattini	C₄	C₂×D₂₅₅
Fitting	C₃×C₅×C₈×C₁₇	C₂
socle	C₂×C₃×C₅×C₁₇	D₄
hypercenter	C₈	D₂₅₅
hypocenter	C₃×C₅×C₁₇	D₈
LCS	G ▷ C₃×C₄×C₅×C₁₇ ▷ C₂×C₃×C₅×C₁₇ ▷ C₃×C₅×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 2040

A minimal faithful permutation representation — 2 generators on 33 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);(2,8)(3,7)(4,6)(10,11)(13,16)(14,15)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

1023 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄₀₈₀, D₆₁₂₀, D₁₀₂₀₀, …
maximal normal subgroup of	D₄₀₈₀
maximal quotient of	D₄₀₈₀, D₆₁₂₀, Dic₂₀₄₀, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 17]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2040 #72211044