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C₉×AGL(1,𝔽₅) #4360

order	180
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×C₃×AGL(1,𝔽₅) (order 120) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)²×AGL(1,ℤ/9)
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

C₉×AGL(1,𝔽₅)

Splittings

Fitting, Hall	C₅×C₉ ⋊ C₄
derived, Sylow	C₅ ⋊ C₄×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₉	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₉
Frattini	C₃	C₃×AGL(1,𝔽₅)
Fitting	C₅×C₉	C₄
socle	C₃×C₅	C₃×C₄
prime radical, cosocle	C₃×D₅	C₂×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₉

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 14 points:

(2,3,4,5)(6,7,9,8,10,12,11,13,14);(1,2,3,5,4)

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×AGL(1,𝔽₅), C₂×C₉×AGL(1,𝔽₅), AGL(1,𝔽₅)×C₂₇, …
maximal normal subgroup of	D₉×AGL(1,𝔽₅), C₂×C₉×AGL(1,𝔽₅), AGL(1,𝔽₅)×C₂₇, …
maximal quotient of	C₉×#170, C₂×C₉×AGL(1,𝔽₅), AGL(1,𝔽₅)×C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 5);