D₂₀ #173

order	40
isoclinism	isoclinic class of Dic₁₀
automorphism	Aut(G) = D₄×AGL(1,𝔽₅) (order 160) · Inn(G) = C₂×D₅ · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = #72213584
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₀
unambiguous semidirect product	C₄⋊D₅

Fitting	C₄×C₅ ⋊ C₂
Sylow	C₅ ⋊ D₄

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 20

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2,3,4)(5,6,7,8,9);(2,4)(6,9)(7,8)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(40, 6);