C₃×#501 #5375

order	192
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	Aut(G) = C₂×#612424 (order 1024) · Inn(G) = C₂×D₄ · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

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

90 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)(11,15)(14,16)(17,18,19);(1,3,4,8)(2,5,6,7)(10,14)(13,16);(9,10,11,13,12,14,15,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 934);