AGL(1,𝔽₅) #52

order	20
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = AGL(1,𝔽₅) (order 20) · Inn(G) = G · Out(G) = C₁
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)²
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

AGL(1,𝔽₅)

C₅ ⋊ C₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅	C₄
prime radical, cosocle	D₅	C₂
LCS	G ▷ C₅
UCS	C₁

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 5
transitive degree	μ_t(G) = 5

A minimal faithful permutation representation — 2 generators on 5 points:

(2,3,4,5);(1,2,3,5,4)

Polycyclic presentation:

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

Here i is the imaginary unit (i² = −1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20, 3);