D₃×C₉ #268

order	54
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×C₃×D₃ (order 36) · Inn(G) = D₃ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = D₃²×AGL(1,ℤ/9)
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₉
unambiguous semidirect product	C₃⋊(C₂×C₉)

Fitting, Sylow	C₃×C₉ ⋊ C₂
derived	C₃ ⋊ C₂×C₉

14 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 4);