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C₃²×AGL(1,𝔽₅) #4375

order	180
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = AGL(1,𝔽₅)×GL(2,𝔽₃) (order 960) · Inn(G) = AGL(1,𝔽₅) · Out(G) = GL(2,𝔽₃)
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

C₃²×AGL(1,𝔽₅)

Splittings

Fitting, socle, Hall	C₃²×C₅ ⋊ C₄
derived, Sylow	C₅ ⋊ C₃²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅	C₄
prime radical	C₃²×D₅	C₂
cosocle	D₅	C₂×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×AGL(1,𝔽₅), C₃⋊D₃×AGL(1,𝔽₅), C₂×C₃²×AGL(1,𝔽₅), …
maximal normal subgroup of	C₃×D₃×AGL(1,𝔽₅), C₃⋊D₃×AGL(1,𝔽₅), C₂×C₃²×AGL(1,𝔽₅), …
maximal quotient of	C₃²×#170, C₂×C₃²×AGL(1,𝔽₅), C₃×C₉×AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 20);