D₉⋊C₉ #4170

order	162
isoclinism	isoclinic class of AGL(1,ℤ/9)
automorphism	Aut(G) = C₃×AGL(1,ℤ/9) (order 162) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C₃
holomorph	G ⋊ Aut(G) = #72243593
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	D₉⋊C₉
unambiguous semidirect product	(C₉⋊C₉)⋊C₂

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

22 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 18 points:

(1,14,17,7,13,12,10,6,11,2,9,18,4,16,5,8,3,15);(1,17,3,10,11,13,4,5,9)(2,14,12,7,16,15,8,6,18)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 6);