C₁₃×S₄ #64631

order	312
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₃×C₄×S₄ (order 288) · Inn(G) = S₄ · Out(G) = C₃×C₄
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₃×S₄
unambiguous semidirect product	A₄⋊(C₂×C₁₃)

Fitting, socle	C₂²×C₁₃ ⋊ D₃
derived	A₄ ⋊ C₂×C₁₃

	subgroup	G/H
center, hypercenter	C₁₃	S₄
commutator, cosocle, hypocenter	A₄	C₂×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₁₃	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₁₃

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 52

A minimal faithful permutation representation — 2 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 47);