C₃⋊(C₃⋊S₄) #6328

order	216
isoclinism	isoclinic class of C₃⋊(C₃⋊S₄)
automorphism	Aut(G) order 93312
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂×C₃³

Aliases

unambiguous semidirect product	C₃⋊(C₃⋊S₄)
unambiguous semidirect product	A₄⋊(C₃⋊D₃)

Fitting, socle	C₂²×C₃² ⋊ D₃
derived	C₃²×A₄ ⋊ C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×A₄	C₂
Fitting, socle	C₂²×C₃²	D₃
LCS	G ▷ C₃²×A₄
UCS	C₁

130 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 4 generators on 10 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 165);