C₃×#400 #5274

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

134 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(5,6)(7,9)(8,10)(11,12)(13,14,16,17)(15,19,20,18)(21,22,23);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,17)(15,20);(2,4)(6,10)(9,12)(13,15,16,20)(14,18,17,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 833);