D₃²×C₁₁ #87689

order	396
isoclinism	isoclinic class of D₃²
automorphism	Aut(G) = C₂×C₅×GO⁺(4,𝔽₂) (order 720) · Inn(G) = D₃² · Out(G) = C₂²×C₅
homology	Abelianization (H₁) = C₂²×C₁₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃²×C₁₁
unambiguous semidirect product	(C₃×D₃)⋊(C₂×C₁₁)
unambiguous semidirect product	(C₃⋊D₃)⋊(C₂×C₁₁)

Fitting, socle, Hall	C₃²×C₁₁ ⋊ C₂²
derived, Sylow	C₃² ⋊ C₂²×C₁₁

	subgroup	G/H
center, hypercenter	C₁₁	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₁₁
Frattini	C₁	G
Fitting, socle	C₃²×C₁₁	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₁₁

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 66

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3)(5,6)(7,8,9,10,11,12,13,14,15,16,17);(2,3)(4,5,6)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(396, 21);