Finite Group Databasebeta

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C₃² #16

order	9
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = GL(2,𝔽₃) (order 48) · Inn(G) = C₁ · Out(G) = GL(2,𝔽₃)
holomorph	G ⋊ Aut(G) = AGL(2,𝔽₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃

Aliases

C₃²

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₁

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 6
transitive degree	μ_t(G) = 9

A minimal faithful permutation representation — 2 generators on 6 points:

(1,2,3);(4,5,6)

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	3	3	3	3	3	3	3	3
χ₁	1	1	1	1	1	1	1	1	1
χ₂	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	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	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	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₅	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$
χ₆	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1
χ₇	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₈	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1
χ₉	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$

Here i is the imaginary unit (i² = −1).

Relationships

center of	C₃²×D₃, C₃²⋊C₉, C₉⋊C₉, …
derived subgroup of	C₃⋊D₃, C₃⋊Dic₃, #157, …
Frattini subgroup of	C₉², C₃²⋊C₉, C₉⋊C₉, …
Fitting subgroup of	C₃×D₃, C₃⋊D₃, #157, …
socle of	C₃×D₃, C₃⋊D₃, C₃×C₉, …
maximal subgroup of	C₃×D₃, C₃⋊D₃, C₂×C₃², …
maximal normal subgroup of	C₃×D₃, C₃⋊D₃, C₂×C₃², …
maximal quotient of	C₂×C₃², C₃×C₉, H(3,𝔽₃), …
outer automorphism group of	#733⋊C₂, D₇×D₉, #4294⋊C₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9, 2);