D₃×AGL(1,ℤ/9) #66426

order	324
isoclinism	isoclinic class of D₃×AGL(1,ℤ/9)
automorphism	Aut(G) = D₃×AGL(1,ℤ/9) (order 324) · Inn(G) = G · Out(G) = C₁
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×AGL(1,ℤ/9)
unambiguous semidirect product	D₉⋊(C₃×D₃)
unambiguous semidirect product	(D₃×D₉)⋊C₃
unambiguous semidirect product	(C₃×AGL(1,ℤ/9))⋊C₂
unambiguous semidirect product	(D₃×3₋¹⁺²)⋊C₂
unambiguous semidirect product	(C₃⋊AGL(1,ℤ/9))⋊C₂

Fitting, Sylow	C₃×3₋¹⁺² ⋊ C₂²
derived	C₃×C₉ ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₉	C₂²×C₃
Frattini	C₃	C₃×D₃²
Fitting	C₃×3₋¹⁺²	C₂²
socle	C₃²	C₂×C₃×D₃
LCS	G ▷ C₃×C₉
UCS	C₁

80 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 118);