C₃⋊AGL(1,ℤ/9) #4206

order	162
isoclinism	isoclinic class of C₃⋊AGL(1,ℤ/9)
automorphism	Aut(G) = #72210336 (order 2916) · Inn(G) = G · Out(G) = C₃⋊D₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₃

Aliases

Fitting, Sylow	C₃×3₋¹⁺² ⋊ C₂
derived	C₃×C₉ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₉	C₂×C₃
Frattini	C₃	C₃×C₃⋊D₃
Fitting	C₃×3₋¹⁺²	C₂
socle	C₃²	C₃×D₃
LCS	G ▷ C₃×C₉
UCS	C₁

49 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:

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

Polycyclic presentation:

2,3^4|c^3=e^2,c^a=c^2e,d^a=d^2,e^a=e^2,c^b=ce

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 42);