C₃×#458 #5332

order	192
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	Aut(G) = C₂²×#1020⋊C₂ (order 768) · Inn(G) = #120 · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂×C₃	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₃
prime radical	C₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ G

160 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 891);