Finite Group Databasebeta

← Prev (#10623599) Next (#10623601) →

D₃₆₀ #10623600

order	720
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

D₃₆₀

Splittings

Fitting	C₅×C₈×C₉ ⋊ C₂
Hall	C₅×C₉ ⋊ D₈
Sylow	C₉ ⋊ D₄₀
Sylow	C₅ ⋊ D₇₂

Subgroups

	subgroup	G/H
center	C₂	D₁₈₀
commutator, cosocle	C₄×C₅×C₉	C₂²
Frattini	C₃×C₄	C₂×D₁₅
Fitting	C₅×C₈×C₉	C₂
socle	C₂×C₃×C₅	D₁₂
hypercenter	C₈	D₄₅
hypocenter	C₅×C₉	D₈
LCS	G ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17)(18,19,20,21,22);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,22)(20,21)

Polycyclic presentation:

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

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#13083694, #13083696, #13083701, …
maximal normal subgroup of	#13083694, #13083696, #13083701, …
maximal quotient of	#13083842, D₇₂₀, Dic₃₆₀, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 69);