Finite Group Databasebeta

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C₂²×#83944 #72246779

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

Splittings

Fitting	C₂³×C₃×AGL(1,ℤ/8) ⋊ C₂
Sylow	C₃ ⋊ C₂²×#3003

Subgroups

	subgroup	G/H
center	C₂⁴	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁶
Frattini	C₂×C₄	C₂⁵×D₃
Fitting	C₂³×C₃×AGL(1,ℤ/8)	C₂
socle	C₂⁴×C₃	C₂²×D₄
hypercenter	C₂³×AGL(1,ℤ/8)	D₃
hypocenter	C₃	C₂²×#3003
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×AGL(1,ℤ/8)

69980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 192

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×D₃×#3003, #72253907, #72254232, …
maximal normal subgroup of	C₂²×D₃×#3003, #72253907, #72254232, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.