C₃×#398 #5272

order	192
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) = C₂²×#6342371 (order 2048) · Inn(G) = C₂³ · Out(G) = C₂³×#120
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

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

150 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 831);