order	144
isoclinism	isoclinic class of Dic₁₈
automorphism	Aut(G) = AGL(1,ℤ/9)×B(4,𝔽₂) (order 3456) · Inn(G) = C₂×D₉ · Out(G) = C₃×#120
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×Dic₁₈
unambiguous semidirect product	(C₂×C₉)⋊Q₈

Splittings

C₉ ⋊ C₂×Q₈

57 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 19 points:

(3,5,6,10)(4,7,8,9)(11,16,18,17,12,14,13,19,15);(1,2)(3,4,6,8)(5,9,10,7)(12,14)(13,17)(15,16)(18,19);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 37);