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H(3,𝔽₃) #81

order	27
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	Aut(G) = AGL(2,𝔽₃) (order 432) · Inn(G) = C₃² · Out(G) = GL(2,𝔽₃)
holomorph	G ⋊ Aut(G) = #72210405
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₃²

Aliases

extraspecial	3₊¹⁺²
unitriangular	UT(3,𝔽₃)
Heisenberg	H(3,𝔽₃)
unambiguous semidirect product	C₃²⋊C₃

Subgroups

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

11 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)(5,9,7);(2,5,6)(4,8,9)

Polycyclic presentation:

3^3|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	3	3	3	3	3	3
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

derived subgroup of	#272, #1084, #1088, …
Fitting subgroup of	(C₃⋊D₃)⋊C₃, #272, #1088, …
maximal subgroup of	(C₃⋊D₃)⋊C₃, #272, C₂×H(3,𝔽₃), …
maximal normal subgroup of	(C₃⋊D₃)⋊C₃, #272, C₂×H(3,𝔽₃), …
maximal quotient of	C₂×H(3,𝔽₃), C₃²⋊C₉, AΣL(1,𝔽₂₇), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27, 3);