A₄⋊#157 #89096

order	432
isoclinism	isoclinic class of A₄⋊#157
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

Fitting, socle	C₂²×C₃² ⋊ Dic₃
derived	C₃²×A₄ ⋊ C₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₃²×A₄	C₄
Fitting, socle	C₂²×C₃²	Dic₃
prime radical, cosocle	A₄×C₃⋊D₃	C₂
LCS	G ▷ C₃²×A₄
UCS	C₁

70 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 10 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 743);