A₄⋊C₃₂ #68005

order	384
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂×C₈×S₄ (order 384) · Inn(G) = S₄ · Out(G) = C₂×C₈
homology	Abelianization (H₁) = C₃₂ · Schur multiplier (H₂) = C₂

Aliases

A₄⋊C₃₂

A₄ ⋊ C₃₂

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	S₄
commutator, hypocenter	A₄	C₃₂
Fitting	C₂²×C₁₆	D₃
socle	C₂³	C₃⋊C₁₆
prime radical	C₂×A₄	C₁₆
cosocle	A₄×C₁₆	C₂
LCS	G ▷ A₄
UCS	C₁ ◁ C₁₆

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

2^5,3,2^2|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^2,g^a=h,h^a=g,g^f=h,h^f=gh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 581);