Dic₄₅ #4358

order	180
isoclinism	isoclinic class of D₄₅
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×AGL(1,ℤ/9) (order 2160) · Inn(G) = D₄₅ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₅
unambiguous semidirect product	C₉⋊Dic₅

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 3);