Finite Group Databasebeta

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B(4,𝔽₂)×AGL(1,𝔽₉) #72253223

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

B(4,𝔽₂)×AGL(1,𝔽₉)

Splittings

Fitting	C₃²×B(4,𝔽₂) ⋊ C₈
Sylow	C₃² ⋊ C₈×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#120×AGL(1,𝔽₉)
commutator	C₂³×C₃²	C₂³×C₈
Frattini	C₂³	C₂³×AGL(1,𝔽₉)
Fitting	C₃²×B(4,𝔽₂)	C₈
socle	C₂×C₃²	C₈×#120
prime radical	C₃⋊D₃×B(4,𝔽₂)	C₄
cosocle	C₂³×#157	C₂⁴
hypercenter	B(4,𝔽₂)	AGL(1,𝔽₉)
hypocenter	C₃²	C₈×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ B(4,𝔽₂)

3311 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 72

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	B(4,𝔽₂)×AΓL(1,𝔽₉)
maximal normal subgroup of	B(4,𝔽₂)×AΓL(1,𝔽₉)

GAP

Past the SmallGroups library coverage — no SmallGroups id.