C₅⋊D₅ #256

order	50
isoclinism	isoclinic class of C₅⋊D₅
automorphism	Aut(G) = AGL(2,𝔽₅) (order 12000) · Inn(G) = G · Out(G) = A₅⋊C₄
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₅

Aliases

C₅⋊D₅

C₅² ⋊ C₂

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

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 25

A minimal faithful permutation representation — 3 generators on 10 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(50, 4);