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(C₃⋊D₃)⋊C₃ #269

order	54
isoclinism	isoclinic class of (C₃⋊D₃)⋊C₃
automorphism	Aut(G) = ((C₃⋊D₃)⋊C₃)⋊C₂ (order 108) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72214399
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product

(C₃⋊D₃)⋊C₃

Splittings

Fitting, Sylow	H(3,𝔽₃) ⋊ C₂
derived	C₃² ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²	C₂×C₃
Frattini, socle	C₃	C₃×D₃
Fitting	H(3,𝔽₃)	C₂
LCS	G ▷ C₃²
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 9 points:

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

Polycyclic presentation:

2,3^3|c^a=c^2,d^a=d^2,c^b=cd

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	2	3	3	6	6	6	9	9	9
order	1	3	3	3	3	3	3	2	6	6
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	-1	-1	-1
χ₃	1	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	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	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	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	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₆	1	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	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	2	2	2	2	-1	-1	-1	0	0	0
χ₈	2	2	$- 1 + i \sqrt{3}$	$- 1 - i \sqrt{3}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	0	0	0
χ₉	2	2	$- 1 - i \sqrt{3}$	$- 1 + i \sqrt{3}$	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	0	0	0
χ₁₀	6	-3	0	0	0	0	0	0	0	0

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

Relationships

maximal subgroup of	((C₃⋊D₃)⋊C₃)⋊C₂, C₂×(C₃⋊D₃)⋊C₃, #4175, …
maximal normal subgroup of	((C₃⋊D₃)⋊C₃)⋊C₂, C₂×(C₃⋊D₃)⋊C₃, C₃×(C₃⋊D₃)⋊C₃, …
maximal quotient of	(C₃⋊Dic₃)⋊C₃, C₂×(C₃⋊D₃)⋊C₃, (C₃⋊D₃)⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 5);