C₃⋊#272 #4210

order	162
automorphism	Aut(G) = #72254100 (order 23328) · Inn(G) = C₃⋊(C₃⋊D₃) · Out(G) = AGL(2,𝔽₃)
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃²

Aliases

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

97 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 4 generators on 12 points:

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

Polycyclic presentation:

2,3^4|b^a=b^2,c^a=c^2,d^a=d^2,d^c=de

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 46);