Finite Group Databasebeta

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D₃×ΓL(1,𝔽₈)×#120 #72244165

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

D₃×ΓL(1,𝔽₈)×#120

Splittings

Fitting	C₃×C₇×#120 ⋊ C₂×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×#120
Sylow	C₇ ⋊ C₃×D₃×#120

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴×C₃
Fitting	C₃×C₇×#120	C₂×C₃
hypercenter	#120	D₃×ΓL(1,𝔽₈)
hypocenter	C₃×C₇	C₂×C₃×#120
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ #120

2778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#120×AGL(1,𝔽₇), C₃⋊D₃×ΓL(1,𝔽₈)×#120, D₃×ΓL(1,𝔽₈)×B(4,𝔽₂), …
maximal normal subgroup of	D₃×#120×AGL(1,𝔽₇), D₃×ΓL(1,𝔽₈)×B(4,𝔽₂), D₃×#89804⋊C₃
maximal quotient of	D₃×ΓL(1,𝔽₈)×#453, C₃⋊D₃×ΓL(1,𝔽₈)×#120, D₃×ΓL(1,𝔽₈)×B(4,𝔽₂)

GAP

Past the SmallGroups library coverage — no SmallGroups id.