C₂⁴⋊C₅ #720

order	80
isoclinism	isoclinic class of C₂⁴⋊C₅
automorphism	Aut(G) = AΓL(1,𝔽₁₆) (order 960) · Inn(G) = G · Out(G) = Dic₃
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₂²

Aliases

C₂⁴⋊C₅

C₂⁴ ⋊ C₅

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂⁴	C₅
LCS	G ▷ C₂⁴
UCS	C₁

17 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 10

A minimal faithful permutation representation — 2 generators on 10 points:

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

Polycyclic presentation:

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

Here ζ is a primitive root of unity of order 5 (ζ⁵ = 1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(80, 49);