C₃×#475 #5349

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) = 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

96 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);(9,10,12,14)(11,16,15,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 908);