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C₉ #15

order	9
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃ (order 6) · Inn(G) = C₁ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/9)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, socle, prime radical, cosocle	C₃	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₃ ▷ C₁

3 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 9

A minimal faithful permutation representation — 1 generator on 9 points:

(1,2,3,4,5,6,7,8,9)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉
size	1	1	1	1	1	1	1	1	1
order	1	9	9	3	9	9	3	9	9
χ₁	1	1	1	1	1	1	1	1	1
χ₂	1	ζ	ζ²	$\frac{- 1 + i \sqrt{3}}{2}$	ζ⁴	ζ⁵	$\frac{- 1 - i \sqrt{3}}{2}$	ζ⁷	ζ⁸
χ₃	1	ζ²	ζ⁴	$\frac{- 1 - i \sqrt{3}}{2}$	ζ⁸	ζ	$\frac{- 1 + i \sqrt{3}}{2}$	ζ⁵	ζ⁷
χ₄	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₅	1	ζ⁴	ζ⁸	$\frac{- 1 + i \sqrt{3}}{2}$	ζ⁷	ζ²	$\frac{- 1 - i \sqrt{3}}{2}$	ζ	ζ⁵
χ₆	1	ζ⁵	ζ	$\frac{- 1 - i \sqrt{3}}{2}$	ζ²	ζ⁷	$\frac{- 1 + i \sqrt{3}}{2}$	ζ⁸	ζ⁴
χ₇	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₈	1	ζ⁷	ζ⁵	$\frac{- 1 + i \sqrt{3}}{2}$	ζ	ζ⁸	$\frac{- 1 - i \sqrt{3}}{2}$	ζ⁴	ζ²
χ₉	1	ζ⁸	ζ⁷	$\frac{- 1 - i \sqrt{3}}{2}$	ζ⁵	ζ⁴	$\frac{- 1 + i \sqrt{3}}{2}$	ζ²	ζ

Here ζ is a primitive root of unity of order 9 (ζ⁹ = 1); i = ζ².

Relationships

center of	D₃×C₉, C₂₇⋊C₃, #737, …
derived subgroup of	D₉, Dic₉, C₂×D₉, …
Frattini subgroup of	C₂₇, D₂₇, C₂×C₂₇, …
Fitting subgroup of	D₉, C₉×A₅, C₉×S₅, …
socle of	#72290071
maximal subgroup of	D₉, C₂×C₉, C₂₇, …
maximal normal subgroup of	D₉, C₂×C₉, C₂₇, …
maximal quotient of	C₂×C₉, C₂₇, C₃×C₉, …
outer automorphism group of	D₁₉, D₂₇, #3866, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9, 1);