C₃×#456 #5330

order	192
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	Aut(G) = C₂²×#2196 (order 512) · Inn(G) = #120 · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂×C₃	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
prime radical	C₃×2₋¹⁺⁴	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ G

142 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 889);