order	130
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₃×C₄×AGL(1,𝔽₅) (order 240) · Inn(G) = D₅ · Out(G) = C₂×C₃×C₄
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₁₃)×D₅⋊AGL(1,𝔽₅)
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₃
unambiguous semidirect product	C₅⋊(C₂×C₁₃)

Splittings

Fitting, socle, Hall	C₅×C₁₃ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₃

	subgroup	G/H
center, hypercenter	C₁₃	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₁₃
Frattini	C₁	G
Fitting, socle	C₅×C₁₃	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₃

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 65

A minimal faithful permutation representation — 2 generators on 18 points:

(2,3)(4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18);(1,2,4,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(130, 1);