D₁₄₄₀ #72210199

order	2880
automorphism	Aut(G) order 552960
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₄₀

Splittings

Fitting	C₅×C₉×C₃₂ ⋊ C₂
Hall	C₅×C₉ ⋊ D₃₂
Sylow	C₉ ⋊ D₁₆₀
Sylow	C₅ ⋊ D₂₈₈

Subgroups

	subgroup	G/H
center	C₂	D₇₂₀
commutator, cosocle	C₅×C₉×C₁₆	C₂²
Frattini	C₃×C₁₆	C₂×D₁₅
Fitting	C₅×C₉×C₃₂	C₂
socle	C₂×C₃×C₅	D₄₈
hypercenter	C₃₂	D₄₅
hypocenter	C₅×C₉	D₃₂
LCS	G ▷ C₅×C₉×C₁₆ ▷ C₅×C₈×C₉ ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 1440

A minimal faithful permutation representation — 2 generators on 46 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);(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)(34,41)(35,40)(36,39)(37,38)(43,46)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

723 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₈₈₀, D₄₃₂₀, D₇₂₀₀, …
maximal normal subgroup of	D₂₈₈₀
maximal quotient of	D₂₈₈₀, D₄₃₂₀, Dic₁₄₄₀, …

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, 3]);;
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> SetPower(coll, 7, F.9);;
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*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1440 #72210199