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AGL(1,ℤ/8)×C₃⋊D₉ #13154377

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

Aliases

AGL(1,ℤ/8)×C₃⋊D₉

Splittings

Fitting	C₃×C₉×AGL(1,ℤ/8) ⋊ C₂
Sylow	C₃×C₉ ⋊ C₂×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄×C₃⋊D₉
commutator, cosocle	C₃×C₄×C₉	C₂⁴
Frattini	C₃×C₄	C₂³×C₃⋊D₃
Fitting	C₃×C₉×AGL(1,ℤ/8)	C₂
socle	C₂×C₃²	C₂×D₃×D₄
hypercenter	AGL(1,ℤ/8)	C₃⋊D₉
hypocenter	C₃×C₉	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

1490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 216

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×AGL(1,ℤ/8)×C₃⋊D₉, AGL(1,ℤ/8)×C₉⋊D₉, AGL(1,ℤ/8)×C₃⋊AGL(1,ℤ/9)
maximal normal subgroup of	C₂×AGL(1,ℤ/8)×C₃⋊D₉, AGL(1,ℤ/8)×C₃⋊AGL(1,ℤ/9)
maximal quotient of	C₃⋊D₉×#442, C₂×AGL(1,ℤ/8)×C₃⋊D₉, AGL(1,ℤ/8)×C₉⋊D₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20440);