A₄×C₄₁ #92452

order	492
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₅×C₈×S₄ (order 960) · Inn(G) = A₄ · Out(G) = C₂×C₅×C₈
homology	Abelianization (H₁) = C₃×C₄₁ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₄₁
unambiguous semidirect product	C₂²⋊(C₃×C₄₁)
unambiguous semidirect product	(C₂²×C₄₁)⋊C₃

Fitting, socle, Hall	C₂²×C₄₁ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₄₁

	subgroup	G/H
center, hypercenter	C₄₁	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₂²×C₄₁	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₄₁

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 164

A minimal faithful permutation representation — 2 generators on 45 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(1,4)(2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(492, 8);