order	234
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₃×C₄×AGL(1,ℤ/9) (order 648) · Inn(G) = D₉ · Out(G) = C₃²×C₄
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×D₉
unambiguous semidirect product	C₉⋊(C₂×C₁₃)

Splittings

Fitting, Hall	C₉×C₁₃ ⋊ C₂
derived, Sylow	C₉ ⋊ C₂×C₁₃

	subgroup	G/H
center, hypercenter	C₁₃	D₉
commutator, cosocle, hypocenter	C₉	C₂×C₁₃
Frattini	C₃	D₃×C₁₃
Fitting	C₉×C₁₃	C₂
socle	C₃×C₁₃	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₁₃

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 117

A minimal faithful permutation representation — 2 generators on 22 points:

(2,4)(3,7)(5,8)(6,9)(10,11,12,13,14,15,16,17,18,19,20,21,22);(1,6,4,3,8,5,7,2,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(234, 3);