C₉×D₅ #775

order	90
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₂×C₃×AGL(1,𝔽₅) (order 120) · Inn(G) = D₅ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/9)×D₅⋊AGL(1,𝔽₅)
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₅
unambiguous semidirect product	C₅⋊(C₂×C₉)

Fitting, Hall	C₅×C₉ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₉

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(90, 2);