D₅⋊C₈ #699

order	80
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = D₄×AGL(1,𝔽₅) (order 160) · Inn(G) = AGL(1,𝔽₅) · Out(G) = D₄
holomorph	G ⋊ Aut(G) = #72221730
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

D₅⋊C₈

C₅ ⋊ C₂×C₈

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 28);