order	78
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₃×C₄×D₃ (order 72) · Inn(G) = D₃ · Out(G) = C₃×C₄
holomorph	G ⋊ Aut(G) = D₃²×AGL(1,𝔽₁₃)
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₁₃
unambiguous semidirect product	C₃⋊(C₂×C₁₃)

Splittings

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

	subgroup	G/H
center, hypercenter	C₁₃	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₁₃

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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(78, 3);