order	250
isoclinism	isoclinic class of C₅⋊D₅
automorphism	Aut(G) = C₄×AGL(2,𝔽₅) (order 48000) · Inn(G) = C₅⋊D₅ · Out(G) = C₄×A₅⋊C₄
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅×C₅⋊D₅
unambiguous semidirect product	C₅⋊(C₅×D₅)

Splittings

Fitting, socle, Sylow	C₅³ ⋊ C₂
derived	C₅² ⋊ C₂×C₅

	subgroup	G/H
center, hypercenter	C₅	C₅⋊D₅
commutator, cosocle, hypocenter	C₅²	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅³	C₂
LCS	G ▷ C₅²
UCS	C₁ ◁ C₅

56 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 50

A minimal faithful permutation representation — 3 generators on 15 points:

(1,2,3,4,5)(7,8)(9,10)(12,13)(14,15);(11,12,14,15,13);(6,7,9,10,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 13);