C₃×#476 #5350

order	192
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	Aut(G) = C₂²×D₄×#120 (order 1024) · Inn(G) = C₂×D₄ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
prime radical	C₂×C₃×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

108 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 909);