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D₃×AΣL(1,𝔽₂₇) #92272

order	486
automorphism	Aut(G) = C₃×D₃×((C₃⋊D₃)⋊C₃)⋊C₂ (order 1944) · Inn(G) = D₃×H(3,𝔽₃) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	D₃×AΣL(1,𝔽₂₇)
unambiguous semidirect product	C₃⋊(C₂×AΣL(1,𝔽₂₇))

Splittings

C₃×AΣL(1,𝔽₂₇) ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	D₃×H(3,𝔽₃)
commutator, cosocle	C₃³	C₂×C₃²
Frattini	C₃²	C₃²×D₃
Fitting	C₃×AΣL(1,𝔽₂₇)	C₂
socle	C₃²	C₂×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	D₃
hypocenter	C₃	C₂×AΣL(1,𝔽₂₇)
LCS	G ▷ C₃³ ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 18

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

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#4174, D₃×#4175, D₃×#4183, …
maximal normal subgroup of	D₃×#4174, D₃×#4175, D₃×#4183, …
maximal quotient of	Dic₃×AΣL(1,𝔽₂₇), C₂×D₃×AΣL(1,𝔽₂₇), D₃×#6856, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 117);