#560⋊C₃ #5449

order	192
isoclinism	isoclinic class of #560⋊C₃
automorphism	Aut(G) = #4626⋊C₂ (order 384) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product	#560⋊C₃
unambiguous semidirect product	(#126⋊C₃)⋊C₂

Fitting, Sylow	#560 ⋊ C₃
derived	C₄² ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₄²	C₂²×C₃
Frattini, socle	C₂²	C₂²×A₄
Fitting	#560	C₃
LCS	G ▷ C₄²
UCS	C₁

67 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 2 generators on 16 points:

(2,12,13,7,3,15)(4,5,8)(6,16,9,10,11,14);(1,15)(2,11)(3,7)(4,9)(5,14)(6,12)(8,13)(10,16)

Polycyclic presentation:

2^2,3,2^4|d^2=fg,e^2=f,d^a=dfg,e^a=ef,d^b=df,e^b=eg,d^c=e,e^c=deg,f^c=g,g^c=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1008);