C₃×#396 #5270

order	192
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 3072
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₃
prime radical	C₂⁴×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

186 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 829);