C₃×#272 #4205

order	162
isoclinism	isoclinic class of #272
automorphism	Aut(G) = D₃×AGL(2,𝔽₃) (order 2592) · Inn(G) = C₃⋊D₃ · Out(G) = D₃×S₄
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

Fitting, Sylow	C₃×H(3,𝔽₃) ⋊ C₂
derived	H(3,𝔽₃) ⋊ C₂×C₃

	subgroup	G/H
center, socle, hypercenter	C₃²	C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂×C₃
Frattini	C₃	C₃×C₃⋊D₃
Fitting	C₃×H(3,𝔽₃)	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃²

76 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 27

A minimal faithful permutation representation — 3 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 41);