A₄⋊C₈ #858

order	96
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂²×S₄ (order 96) · Inn(G) = S₄ · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = #72258669
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₂

Aliases

A₄⋊C₈

A₄ ⋊ C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	S₄
commutator, hypocenter	A₄	C₈
Fitting	C₂²×C₄	D₃
socle	C₂³	Dic₃
prime radical	C₂×A₄	C₄
cosocle	C₄×A₄	C₂
LCS	G ▷ A₄
UCS	C₁ ◁ C₄

26 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

(1,2,4,3)(5,6,7,9,8,10,11,12);(2,3,4)

Polycyclic presentation:

2^3,3,2^2|a^2=b,b^2=c,d^a=d^2,e^a=f,f^a=e,e^d=f,f^d=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 65);