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C₃⋊D₃×H(3,𝔽₃) #92386

order	486
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃³

Aliases

direct product	C₃⋊D₃×H(3,𝔽₃)
unambiguous semidirect product	C₃⋊(D₃×H(3,𝔽₃))

Splittings

C₃²×H(3,𝔽₃) ⋊ C₂

Subgroups

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

324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 15 points:

(2,3)(5,6)(8,11,12)(10,14,15);(4,5,6)(7,8,10)(9,12,14)(11,15,13);(1,2,3)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

derived subgroup of	D₃²⋊#272, #14519127, #14519133, …
maximal subgroup of	H(3,𝔽₃)×#157, (C₃⋊(C₃⋊#157))⋊C₃, #11758047, …
maximal normal subgroup of	H(3,𝔽₃)×#157, (C₃⋊(C₃⋊#157))⋊C₃, #11758047, …
maximal quotient of	H(3,𝔽₃)×C₃⋊Dic₃, C₂×C₃⋊D₃×H(3,𝔽₃), C₃⋊D₃×C₃²⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 231);