Finite Group Databasebeta

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D₃×#6500⋊C₃ #72244183

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

Aliases

D₃×#6500⋊C₃

Splittings

Fitting	C₂×C₃×C₇×D₄ ⋊ C₂²×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×#120
Sylow	C₇ ⋊ C₃×D₃×#120

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴×C₃
Fitting	C₂×C₃×C₇×D₄	C₂²×C₃
hypercenter	C₂×D₄	D₃×AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₂×C₃×#120
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

2778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#120×AGL(1,𝔽₇), C₃⋊D₃×#6500⋊C₃, D₃×#13052144, …
maximal normal subgroup of	D₃×#120×AGL(1,𝔽₇), D₃×#13052144, D₃×#89513⋊C₃
maximal quotient of	D₃×#89970⋊C₃, D₃×#89849⋊C₃, C₃⋊D₃×#6500⋊C₃, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.