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C₃×AΓL(1,𝔽₉) #89090

order	432
isoclinism	isoclinic class of AΓL(1,𝔽₉)
automorphism	Aut(G) = C₂×AΓL(1,𝔽₉) (order 288) · Inn(G) = AΓL(1,𝔽₉) · Out(G) = C₂
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×AΓL(1,𝔽₉)
unambiguous semidirect product	AGL(1,𝔽₉)⋊(C₂×C₃)

Splittings

Fitting, socle, Sylow

C₃³ ⋊ SD₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₃	AΓL(1,𝔽₉)
commutator, cosocle	#157	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃³	SD₁₆
prime radical	C₃×GO⁺(4,𝔽₂)	C₂
hypocenter	C₃²	C₃×SD₁₆
LCS	G ▷ #157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₃

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AΓL(1,𝔽₉), C₂×C₃×AΓL(1,𝔽₉), C₉×AΓL(1,𝔽₉), …
maximal normal subgroup of	D₃×AΓL(1,𝔽₉), C₂×C₃×AΓL(1,𝔽₉), C₉×AΓL(1,𝔽₉), …
maximal quotient of	C₃×GO⁺(4,𝔽₂)⋊C₄, C₃×PSU(3,𝔽₂)⋊C₄, C₃×AGL(1,𝔽₉)⋊C₄, …
automorphism group of	#66419

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 737);