Finite Group Databasebeta

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C₂×C₃⋊D₃×(C₂²⋊D₇)⋊C₃ #72245698

order	6048
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

C₂×C₃⋊D₃×(C₂²⋊D₇)⋊C₃

Splittings

Fitting	C₂³×C₃²×C₇ ⋊ C₂²×C₃
Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₂²×D₄
Sylow	C₇ ⋊ C₂×C₃×D₄×C₃⋊D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₃⋊D₃×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₃²×C₇	C₂⁴×C₃
Frattini	C₂	C₂²×C₃⋊D₃×AGL(1,𝔽₇)
Fitting	C₂³×C₃²×C₇	C₂²×C₃
socle	C₂²×C₃²×C₇	C₂³×C₃
hypercenter	C₂³	C₃⋊D₃×AGL(1,𝔽₇)
hypocenter	C₃²×C₇	C₂²×C₃×D₄
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

3264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 504

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₄×C₃⋊D₃×AGL(1,𝔽₇), C₃⋊D₃×#6503⋊C₃, C₃⋊D₃×#6500⋊C₃, …
maximal normal subgroup of	C₂×D₄×C₃⋊D₃×AGL(1,𝔽₇), C₃⋊D₃×#6503⋊C₃, C₃⋊D₃×#6500⋊C₃, …
maximal quotient of	C₃⋊D₃×#6503⋊C₃, C₃⋊D₃×#6500⋊C₃, C₃⋊D₃×#10620789

GAP

Past the SmallGroups library coverage — no SmallGroups id.