Finite Group Databasebeta

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C₂³×C₃⋊D₃×AGL(1,𝔽₇) #72245685

order	6048
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂¹⁰×C₃

Aliases

C₂³×C₃⋊D₃×AGL(1,𝔽₇)

Splittings

Fitting, socle	C₂³×C₃²×C₇ ⋊ C₂²×C₃
derived	C₃²×C₇ ⋊ C₂⁵×C₃
Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₂⁵
Sylow	C₇ ⋊ C₂⁴×C₃×C₃⋊D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	C₃⋊D₃×AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₃²×C₇	C₂⁵×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₃²×C₇	C₂²×C₃
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₂³

10048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 504

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₄×C₃⋊D₃×AGL(1,𝔽₇), C₂×AGL(1,𝔽₇)×#3811, C₂⁴×C₃⋊D₃×AGL(1,𝔽₇), …
maximal normal subgroup of	C₂×D₄×C₃⋊D₃×AGL(1,𝔽₇), C₂×AGL(1,𝔽₇)×#3811, C₂⁴×C₃⋊D₃×AGL(1,𝔽₇), …
maximal quotient of	C₂×D₄×C₃⋊D₃×AGL(1,𝔽₇), C₂⁴×C₃⋊D₃×AGL(1,𝔽₇)

GAP

Past the SmallGroups library coverage — no SmallGroups id.