order	186
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×C₃×C₅×D₃ (order 180) · Inn(G) = D₃ · Out(G) = C₂×C₃×C₅
holomorph	G ⋊ Aut(G) = D₃²×AGL(1,𝔽₃₁)
homology	Abelianization (H₁) = C₂×C₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₃₁
unambiguous semidirect product	C₃⋊(C₂×C₃₁)

Splittings

Fitting, socle, Hall	C₃×C₃₁ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₃₁

	subgroup	G/H
center, hypercenter	C₃₁	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₃₁
Frattini	C₁	G
Fitting, socle	C₃×C₃₁	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₃₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 93

A minimal faithful permutation representation — 2 generators on 34 points:

(2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34);(1,2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(186, 3);