order	192
isoclinism	isoclinic class of #120⋊C₃
automorphism	Aut(G) = C₂×#4625⋊C₂ (order 768) · Inn(G) = #120⋊C₃ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×#120⋊C₃
unambiguous semidirect product	(C₂×#120)⋊C₃

Splittings

Fitting, Sylow	C₂×#120 ⋊ C₃
derived	C₂⁴ ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₂	#120⋊C₃
commutator, cosocle, hypocenter	C₂⁴	C₂²×C₃
Frattini	C₂²	C₂²×A₄
Fitting	C₂×#120	C₃
socle	C₂³	C₂×A₄
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₂

142 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 12

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1000);