A₄⋊C₁₆ #4627

order	192
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂×C₄×S₄ (order 192) · Inn(G) = S₄ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₂

Aliases

A₄⋊C₁₆

A₄ ⋊ C₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₈	S₄
commutator, hypocenter	A₄	C₁₆
Fitting	C₂²×C₈	D₃
socle	C₂³	C₃⋊C₈
prime radical	C₂×A₄	C₈
cosocle	C₈×A₄	C₂
LCS	G ▷ A₄
UCS	C₁ ◁ C₈

33 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 48

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

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

Polycyclic presentation:

2^4,3,2^2|a^2=b,b^2=c,c^2=d,e^a=e^2,f^a=g,g^a=f,f^e=g,g^e=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 186);