order	144
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂×C₃×AGL(1,ℤ/8) (order 192) · Inn(G) = D₄ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₁

Aliases

C₉×Q₁₆

Subgroups

27 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17)(18,25,24,23,22,21,20,19);(10,18,14,22)(11,19,15,23)(12,20,16,24)(13,21,17,25)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 27);