C₁₇×S₄ #87999

order	408
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₁₆×S₄ (order 384) · Inn(G) = S₄ · Out(G) = C₁₆
homology	Abelianization (H₁) = C₂×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₇×S₄
unambiguous semidirect product	A₄⋊(C₂×C₁₇)

Fitting, socle	C₂²×C₁₇ ⋊ D₃
derived	A₄ ⋊ C₂×C₁₇

	subgroup	G/H
center, hypercenter	C₁₇	S₄
commutator, cosocle, hypocenter	A₄	C₂×C₁₇
Frattini	C₁	G
Fitting, socle	C₂²×C₁₇	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₁₇

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 68

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(408, 36);