order	80
isoclinism	isoclinic class of #700
automorphism	Aut(G) = D₄×AGL(1,𝔽₅) (order 160) · Inn(G) = C₂×AGL(1,𝔽₅) · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Fitting	C₄×C₅ ⋊ C₄
Sylow	C₅ ⋊ C₄⋊C₄

26 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 31);