C₂×C₃×D₃×C₁₁ #87694

order	396
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂³×C₅×D₃ (order 240) · Inn(G) = D₃ · Out(G) = C₂³×C₅
homology	Abelianization (H₁) = C₂²×C₃×C₁₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃×D₃×C₁₁
unambiguous semidirect product	C₃⋊(C₂²×C₃×C₁₁)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₃×C₁₁)

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₁	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₃×C₁₁
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₁₁	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₁₁

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 132

A minimal faithful permutation representation — 2 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(396, 26);