order	130
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) = C₄×AGL(1,𝔽₁₃) (order 624) · Inn(G) = D₁₃ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₃
unambiguous semidirect product	C₁₃⋊(C₂×C₅)

Splittings

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

	subgroup	G/H
center, hypercenter	C₅	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₅

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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(130, 2);