D₉ #44

order	18
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = AGL(1,ℤ/9) (order 54) · Inn(G) = G · Out(G) = C₃
holomorph	G ⋊ Aut(G) = #92265⋊C₂
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉
unambiguous semidirect product	C₉⋊C₂

C₉ ⋊ C₂

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉	C₂
Frattini, socle	C₃	D₃
LCS	G ▷ C₉
UCS	C₁

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 9

A minimal faithful permutation representation — 2 generators on 9 points:

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

2,3^2|b^3=c,b^a=b^2c^2,c^a=c^2

Here ζ is a primitive root of unity of order 9 (ζ⁹ = 1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18, 1);