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3₋¹⁺² #82

order	27
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	Aut(G) = #272 (order 54) · Inn(G) = C₃² · Out(G) = D₃
holomorph	G ⋊ Aut(G) = #13091120
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₁

Aliases

extraspecial	3₋¹⁺²
unambiguous semidirect product	C₉⋊C₃

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃	C₃²
prime radical	C₃²	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ G
LECS	G ▷ C₃ ▷ C₁

8 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:

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

Polycyclic presentation:

3^3|a^3=c,b^a=bc

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	1	1	3	3	3	3	3	3	3	3
order	1	3	3	3	3	9	9	9	9	9	9
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	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	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	1	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	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	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	1	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	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$
χ₁₀	3	$\frac{- 3 + 3 i \sqrt{3}}{2}$	$\frac{- 3 - 3 i \sqrt{3}}{2}$	0	0	0	0	0	0	0	0
χ₁₁	3	$\frac{- 3 - 3 i \sqrt{3}}{2}$	$\frac{- 3 + 3 i \sqrt{3}}{2}$	0	0	0	0	0	0	0	0

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

Relationships

Fitting subgroup of	AGL(1,ℤ/9), 3₋¹⁺²×A₅
maximal subgroup of	AGL(1,ℤ/9), C₂×3₋¹⁺², AΣL(1,𝔽₂₇), …
maximal normal subgroup of	AGL(1,ℤ/9), C₂×3₋¹⁺², AΣL(1,𝔽₂₇), …
maximal quotient of	C₂×3₋¹⁺², C₃²⋊C₉, C₉⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27, 4);