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3₋¹⁺²⋊(C₂²⋊C₉) #11757457

order	972
automorphism	?
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₂×C₃

Aliases

unambiguous semidirect product

3₋¹⁺²⋊(C₂²⋊C₉)

Splittings

C₂² ⋊ 3₋¹⁺²⋊C₉

Subgroups

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator	C₂²×C₃²	C₃×C₉
Frattini	C₃³	C₃×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
prime radical, cosocle	C₂²×C₃³	C₃²
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	3₋¹⁺²⋊C₉
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 162

Polycyclic presentation:

3^5,2^2|a^3=c,b^3=e,b^a=bd,f^a=fg,g^a=f,d^b=de

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

derived subgroup of	(3₋¹⁺²⋊(C₂²⋊C₉))⋊C₂
maximal subgroup of	(3₋¹⁺²⋊(C₂²⋊C₉))⋊C₂, C₂×3₋¹⁺²⋊(C₂²⋊C₉)
maximal normal subgroup of	(3₋¹⁺²⋊(C₂²⋊C₉))⋊C₂, C₂×3₋¹⁺²⋊(C₂²⋊C₉)
maximal quotient of	3₋¹⁺²⋊(Q₈⋊C₉), C₂×3₋¹⁺²⋊(C₂²⋊C₉)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(972, 182);