C₅×D₅ #255

order	50
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₄×AGL(1,𝔽₅) (order 80) · Inn(G) = D₅ · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×D₅⋊AGL(1,𝔽₅)
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

affine special semilinear	AΣL(1,𝔽₂₅)
direct product	C₅×D₅
unambiguous semidirect product	C₅⋊(C₂×C₅)

Fitting, socle, Sylow	C₅² ⋊ C₂
derived	C₅ ⋊ C₂×C₅

	subgroup	G/H
center, hypercenter	C₅	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅²	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₅

10 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(50, 3);