D₆₅ #3601

order	130
isoclinism	isoclinic class of D₆₅
automorphism	Aut(G) = AGL(1,𝔽₅)×AGL(1,𝔽₁₃) (order 3120) · Inn(G) = G · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₅
unambiguous semidirect product	C₅⋊D₁₃
unambiguous semidirect product	C₁₃⋊D₅

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

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

Polycyclic presentation:

2,5,13|b^a=b^4,c^a=c^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(130, 3);