C₁₉×S₄ #90722

order	456
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂×C₉×S₄ (order 432) · Inn(G) = S₄ · Out(G) = C₂×C₉
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₉×S₄
unambiguous semidirect product	A₄⋊(C₂×C₁₉)

Fitting, socle	C₂²×C₁₉ ⋊ D₃
derived	A₄ ⋊ C₂×C₁₉

	subgroup	G/H
center, hypercenter	C₁₉	S₄
commutator, cosocle, hypocenter	A₄	C₂×C₁₉
Frattini	C₁	G
Fitting, socle	C₂²×C₁₉	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₁₉

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 76

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23);(20,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 42);