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(C₃⋊D₃)⋊C₃×B(4,𝔽₂) #72242765

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷×C₃

Aliases

(C₃⋊D₃)⋊C₃×B(4,𝔽₂)

Splittings

Fitting	H(3,𝔽₃)×B(4,𝔽₂) ⋊ C₂
Sylow	H(3,𝔽₃) ⋊ C₂×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#120×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂³×C₃²	C₂⁴×C₃
Frattini	C₂³×C₃	C₂³×C₃×D₃
Fitting	H(3,𝔽₃)×B(4,𝔽₂)	C₂
socle	C₂×C₃	C₃×D₃×#120
hypercenter	B(4,𝔽₂)	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ B(4,𝔽₂)

3761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	B(4,𝔽₂)×PSU(3,𝔽₂)⋊C₃, B(4,𝔽₂)×((C₃⋊D₃)⋊C₃)⋊C₂
maximal normal subgroup of	B(4,𝔽₂)×((C₃⋊D₃)⋊C₃)⋊C₂

GAP

Past the SmallGroups library coverage — no SmallGroups id.