order	78
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₃) (order 312) · Inn(G) = D₁₃ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = D₃×#72210302
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

C₃×D₁₃

Splittings

Fitting, socle, Hall	C₃×C₁₃ ⋊ C₂
derived, Sylow	C₁₃ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₃	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 39

A minimal faithful permutation representation — 2 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(78, 4);