Finite Group Databasebeta

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C₂²×#87192 #72271221

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

Splittings

Fitting	C₂³×C₃×AGL(1,ℤ/8) ⋊ C₂
Sylow	C₃ ⋊ C₂²×#3585

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁷
Frattini	C₄	C₂⁶×D₃
Fitting	C₂³×C₃×AGL(1,ℤ/8)	C₂
socle	C₂³×C₃	C₂³×D₄
hypercenter	C₂³×AGL(1,ℤ/8)	D₃
hypocenter	C₃	C₂²×#3585
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×AGL(1,ℤ/8)

115668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id.