C₂×C₃×D₃×C₁₃ #90876

order	468
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂²×C₃×C₄×D₃ (order 288) · Inn(G) = D₃ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₂²×C₃×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃×D₃×C₁₃
unambiguous semidirect product	C₃⋊(C₂²×C₃×C₁₃)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₃×C₁₃)

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₃	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₁₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₁₃

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 2 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(468, 51);