C₃×#500 #5374

order	192
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	Aut(G) = C₂×#72215640 (order 4096) · Inn(G) = C₂×D₄ · Out(G) = C₂×#2846
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

98 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2,4,6)(3,7,8,5)(10,13)(11,14)(15,16)(17,18,19);(1,3,4,8)(2,5,6,7);(9,10,14,16,12,15,11,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 933);