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B(4,𝔽₂)×ΓL(1,𝔽₈)⋊D₇ #72246192

order	18816
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

B(4,𝔽₂)×ΓL(1,𝔽₈)⋊D₇

Splittings

Fitting	C₇²×B(4,𝔽₂) ⋊ C₂×C₃
Hall	B(4,𝔽₂)×C₇⋊D₇ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×B(4,𝔽₂)
Sylow	C₇² ⋊ C₂×C₃×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#120×ΓL(1,𝔽₈)⋊D₇
commutator, cosocle	C₂³×C₇²	C₂⁴×C₃
Frattini	C₂³	C₂³×ΓL(1,𝔽₈)⋊D₇
Fitting	C₇²×B(4,𝔽₂)	C₂×C₃
socle	C₂×C₇²	C₂×C₃×#120
hypercenter	B(4,𝔽₂)	ΓL(1,𝔽₈)⋊D₇
hypocenter	C₇²	C₂×C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ B(4,𝔽₂)

5090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	B(4,𝔽₂)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
maximal normal subgroup of	B(4,𝔽₂)×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)

GAP

Past the SmallGroups library coverage — no SmallGroups id.