D₄₅ #776

order	90
isoclinism	isoclinic class of D₄₅
automorphism	Aut(G) = AGL(1,𝔽₅)×AGL(1,ℤ/9) (order 1080) · Inn(G) = G · Out(G) = C₃×C₄
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₄₅
unambiguous semidirect product	C₅⋊D₉
unambiguous semidirect product	C₉⋊D₅

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₉	C₂
Frattini	C₃	D₁₅
socle	C₃×C₅	D₃
LCS	G ▷ C₅×C₉
UCS	C₁

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:

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

Polycyclic presentation:

2,3,5,3|b^3=d^2,b^a=b^2d,c^a=c^4,d^a=d^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(90, 3);