C₃×#487 #5361

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) = C₄

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

98 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,19);(1,3,4,7)(2,5,6,8)(9,10,11,13,12,14,16,15);(2,6)(5,8)(9,10,11,13,12,14,16,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 920);