Finite Group Databasebeta

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B(4,𝔽₂)×((C₃⋊D₃)⋊C₃)⋊C₂ #72253279

order	6912
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

B(4,𝔽₂)×((C₃⋊D₃)⋊C₃)⋊C₂

Splittings

Fitting	H(3,𝔽₃)×B(4,𝔽₂) ⋊ C₂²
Sylow	H(3,𝔽₃) ⋊ C₂²×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#120×((C₃⋊D₃)⋊C₃)⋊C₂
commutator, cosocle	C₂³×H(3,𝔽₃)	C₂⁵
Frattini	C₂³×C₃	C₂³×D₃²
Fitting	H(3,𝔽₃)×B(4,𝔽₂)	C₂²
socle	C₂×C₃	D₃²×#120
hypercenter	B(4,𝔽₂)	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	C₂²×B(4,𝔽₂)
LCS	G ▷ C₂³×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ B(4,𝔽₂)

25567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

B(4,𝔽₂)×AGL(2,𝔽₃)

GAP

Past the SmallGroups library coverage — no SmallGroups id.