order	198
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂×C₅×AGL(2,𝔽₃) (order 4320) · Inn(G) = C₃⋊D₃ · Out(G) = C₂×C₅×S₄
homology	Abelianization (H₁) = C₂×C₁₁ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₁₁×C₃⋊D₃
unambiguous semidirect product	C₃⋊(D₃×C₁₁)

Splittings

Fitting, socle, Hall	C₃²×C₁₁ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₁₁

	subgroup	G/H
center, hypercenter	C₁₁	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂×C₁₁
Frattini	C₁	G
Fitting, socle	C₃²×C₁₁	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₁₁

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 99

A minimal faithful permutation representation — 3 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(198, 8);