C₃×#472 #5346

order	192
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	Aut(G) = C₂×D₄×AGL(1,ℤ/8) (order 512) · Inn(G) = #121 · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂×C₃	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ G

108 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 905);