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C₃×C₃⋊D₃×AGL(1,ℤ/8) #13179523

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

C₃×C₃⋊D₃×AGL(1,ℤ/8)

Splittings

Fitting	C₃³×AGL(1,ℤ/8) ⋊ C₂
Sylow	C₃³ ⋊ C₂×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂×D₄×C₃⋊D₃
commutator, cosocle	C₃²×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×AGL(1,ℤ/8)	C₂
socle	C₂×C₃³	C₂²×D₄
hypercenter	C₃×AGL(1,ℤ/8)	C₃⋊D₃
hypocenter	C₃²	C₂×C₃×AGL(1,ℤ/8)
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×AGL(1,ℤ/8)

2060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 144

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×C₃⋊D₃×AGL(1,ℤ/8), AGL(1,ℤ/8)×C₃⋊AGL(1,ℤ/9), C₂×C₃×C₃⋊D₃×AGL(1,ℤ/8)
maximal normal subgroup of	D₃×C₃⋊D₃×AGL(1,ℤ/8), C₂×C₃×C₃⋊D₃×AGL(1,ℤ/8)
maximal quotient of	C₃×C₃⋊D₃×#442, AGL(1,ℤ/8)×C₃⋊AGL(1,ℤ/9), C₂×C₃×C₃⋊D₃×AGL(1,ℤ/8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 45586);