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C₂²⋊C₉ #151

order	36
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃×S₄ (order 72) · Inn(G) = A₄ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72213207
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

C₂²⋊C₉

Splittings

C₂² ⋊ C₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	A₄
commutator, hypocenter	C₂²	C₉
Fitting, socle, prime radical, cosocle	C₂²×C₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 13 points:

(2,3,4)(5,6,8,7,9,11,10,12,13);(1,4)(2,3)

Polycyclic presentation:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	1	1	3	3	3	4	4	4	4	4	4
order	1	3	3	2	6	6	9	9	9	9	9	9
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	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	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	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	ζ	ζ⁴	ζ⁷	ζ²	ζ⁵	ζ⁸
χ₅	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	ζ⁴	ζ⁷	ζ	ζ⁸	ζ²	ζ⁵
χ₆	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	ζ⁷	ζ	ζ⁴	ζ⁵	ζ⁸	ζ²
χ₇	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	ζ²	ζ⁸	ζ⁵	ζ⁴	ζ	ζ⁷
χ₈	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	ζ⁵	ζ²	ζ⁸	ζ	ζ⁷	ζ⁴
χ₉	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	ζ⁸	ζ⁵	ζ²	ζ⁷	ζ⁴	ζ
χ₁₀	3	3	3	-1	-1	-1	0	0	0	0	0	0
χ₁₁	3	$\frac{- 3 + 3 i \sqrt{3}}{2}$	$\frac{- 3 - 3 i \sqrt{3}}{2}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	0	0	0	0	0	0
χ₁₂	3	$\frac{- 3 - 3 i \sqrt{3}}{2}$	$\frac{- 3 + 3 i \sqrt{3}}{2}$	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	0	0	0	0	0	0

Here ζ is a primitive root of unity of order 9 (ζ⁹ = 1); i = ζ².

Relationships

derived subgroup of	(C₂²⋊C₉)⋊C₂, (C₂²⋊C₉)⋊C₄, C₂×(C₂²⋊C₉)⋊C₂, …
maximal subgroup of	(C₂²⋊C₉)⋊C₂, C₂×C₂²⋊C₉, C₉×A₄, …
maximal normal subgroup of	(C₂²⋊C₉)⋊C₂, C₂×C₂²⋊C₉, C₉×A₄, …
maximal quotient of	Q₈⋊C₉, C₂×C₂²⋊C₉, C₂²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36, 3);