Finite Group Databasebeta

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#272

order	54
isoclinism	isoclinic class of #272
automorphism	Aut(G) = AGL(2,𝔽₃) (order 432) · Inn(G) = C₃⋊D₃ · Out(G) = S₄
holomorph	G ⋊ Aut(G) = #72214644
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(3,𝔽₃) ⋊ C₂

Subgroups

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

22 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 9 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	1	1	6	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	2	2	2	2	-1	-1	-1	0	0	0
χ₄	2	2	2	-1	2	-1	-1	0	0	0
χ₅	2	2	2	-1	-1	2	-1	0	0	0
χ₆	2	2	2	-1	-1	-1	2	0	0	0
χ₇	3	$\frac{- 3 + 3 i \sqrt{3}}{2}$	$\frac{- 3 - 3 i \sqrt{3}}{2}$	0	0	0	0	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₈	3	$\frac{- 3 + 3 i \sqrt{3}}{2}$	$\frac{- 3 - 3 i \sqrt{3}}{2}$	0	0	0	0	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₉	3	$\frac{- 3 - 3 i \sqrt{3}}{2}$	$\frac{- 3 + 3 i \sqrt{3}}{2}$	0	0	0	0	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₁₀	3	$\frac{- 3 - 3 i \sqrt{3}}{2}$	$\frac{- 3 + 3 i \sqrt{3}}{2}$	0	0	0	0	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$

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

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 8);