C₃×A₄×C₁₃ #90873

order	468
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₃×C₄×D₃×S₄ (order 1728) · Inn(G) = A₄ · Out(G) = C₂×C₃×C₄×D₃
homology	Abelianization (H₁) = C₃²×C₁₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃×A₄×C₁₃
unambiguous semidirect product	C₂²⋊(C₃²×C₁₃)
unambiguous semidirect product	(C₂²×C₃)⋊(C₃×C₁₃)

	subgroup	G/H
center, hypercenter	C₃×C₁₃	A₄
commutator, cosocle, hypocenter	C₂²	C₃²×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₁₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₁₃

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 2 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(468, 48);