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C₃×D₃ #46

order	18
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃ (order 12) · Inn(G) = D₃ · Out(G) = C₂
holomorph	G ⋊ Aut(G) = D₃³
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

general unitary	GU(2,𝔽₂)
affine special semilinear	AΣL(1,𝔽₉)
direct product	C₃×D₃
unambiguous semidirect product	C₃⋊(C₂×C₃)

Splittings

Fitting, socle, Sylow	C₃² ⋊ C₂
derived	C₃ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃²	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₃

9 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉
size	1	2	3	1	2	3	1	2	3
order	1	3	2	3	3	6	3	3	6
χ₁	1	1	1	1	1	1	1	1	1
χ₂	1	1	-1	1	1	-1	1	1	-1
χ₃	2	-1	0	2	-1	0	2	-1	0
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	1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₆	2	-1	0	$- 1 + i \sqrt{3}$	$\frac{1 - i \sqrt{3}}{2}$	0	$- 1 - i \sqrt{3}$	$\frac{1 + i \sqrt{3}}{2}$	0
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	1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₉	2	-1	0	$- 1 - i \sqrt{3}$	$\frac{1 + i \sqrt{3}}{2}$	0	$- 1 + i \sqrt{3}$	$\frac{1 - i \sqrt{3}}{2}$	0

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

Relationships

maximal subgroup of	D₃², C₂×C₃×D₃, C₃×D₉, …
maximal normal subgroup of	D₃², C₂×C₃×D₃, D₃×C₉, …
maximal quotient of	C₃×Dic₃, C₂×C₃×D₃, C₃×D₉, …
outer automorphism group of	C₂₇⋊C₃, #4185, C₂×C₂₇⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18, 3);