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C₄×D₃×H(3,𝔽₃) #10619930

order	648
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product	C₄×D₃×H(3,𝔽₃)
unambiguous semidirect product	(C₃²×C₄×D₃)⋊C₃

Splittings

Fitting	C₃×C₄×H(3,𝔽₃) ⋊ C₂
Sylow	C₃×H(3,𝔽₃) ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center	C₃×C₄	C₃²×D₃
commutator	C₃²	C₂×C₃²×C₄
Frattini	C₂×C₃	C₂×C₃²×D₃
Fitting	C₃×C₄×H(3,𝔽₃)	C₂
socle, cosocle	C₂×C₃²	C₂²×C₃²
prime radical	C₂×D₃×H(3,𝔽₃)	C₂
hypercenter	C₄×H(3,𝔽₃)	D₃
hypocenter	C₃	C₂×C₄×H(3,𝔽₃)
LCS	G ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₄×H(3,𝔽₃)

227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×(C₃⋊(C₃⋊C₈))⋊C₃, #88786⋊C₃, D₃×#6180, …
maximal normal subgroup of	D₃×(C₃⋊(C₃⋊C₈))⋊C₃, #88786⋊C₃, D₃×#6180, …
maximal quotient of	D₃×C₈×H(3,𝔽₃), H(3,𝔽₃)×ΓL(1,𝔽₂₅), C₄×Dic₃×H(3,𝔽₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 449);