C₃×#395 #5269

order	192
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 6144
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

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

182 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)(9,10,13,15)(11,16,12,14)(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, 828);