order	234
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₃×C₄×AGL(2,𝔽₃) (order 5184) · Inn(G) = C₃⋊D₃ · Out(G) = C₃×C₄×S₄
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₃

Aliases

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

Splittings

Fitting, socle, Hall	C₃²×C₁₃ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₁₃

	subgroup	G/H
center, hypercenter	C₁₃	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂×C₁₃
Frattini	C₁	G
Fitting, socle	C₃²×C₁₃	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₁₃

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 117

A minimal faithful permutation representation — 3 generators on 19 points:

(2,3)(5,6)(7,8,9,10,11,12,13,14,15,16,17,18,19);(4,5,6);(1,2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(234, 14);