C₂×C₃×H(3,𝔽₃) #4212

order	162
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	Aut(G) = #72222023 (order 23328) · Inn(G) = C₃² · Out(G) = D₃×AGL(2,𝔽₃)
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product	C₂×C₃×H(3,𝔽₃)
unambiguous semidirect product	(C₂×C₃²)⋊C₃²

	subgroup	G/H
center, socle	C₂×C₃²	C₃²
commutator, Frattini, cosocle	C₃	C₂×C₃³
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ G

112 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 48);