C₃×#486 #5360

order	192
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	Aut(G) = C₂×#72210469 (order 2048) · Inn(G) = C₂×D₄ · Out(G) = C₂²×#120
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
prime radical	C₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

118 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)(3,5)(4,6)(7,8)(10,11)(12,13)(15,16)(17,18,19);(1,3,4,7)(2,5,6,8)(9,10,13,15,14,16,12,11);(2,6)(5,8)(9,11,12,16,14,15,13,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 919);