C₃⋊(C₅⋊S₄) #67211

order	360
isoclinism	isoclinic class of C₃⋊(C₅⋊S₄)
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂×C₃

Aliases

unambiguous semidirect product	C₃⋊(C₅⋊S₄)
unambiguous semidirect product	C₅⋊(C₃⋊S₄)
unambiguous semidirect product	A₄⋊D₁₅

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₅×A₄	C₂
Fitting, socle	C₂²×C₃×C₅	D₃
LCS	G ▷ C₃×C₅×A₄
UCS	C₁

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 3 generators on 12 points:

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

Polycyclic presentation:

2,3^2,5,2^2|b^a=b^2,c^a=c^2,d^a=d^4,e^a=f,f^a=e,e^b=ef,f^b=e

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 141);