C₉⋊C₈ #604

order	72
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂²×AGL(1,ℤ/9) (order 216) · Inn(G) = D₉ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = #72219937
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

C₉⋊C₈

C₉ ⋊ C₈

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 1);