C₃×#490 #5364

order	192
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	Aut(G) = C₂×#72213256 (order 2048) · Inn(G) = C₂×D₄ · Out(G) = C₂×B(4,𝔽₂)
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = 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

110 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)(10,15)(11,16)(12,13)(17,18,19);(1,3,4,7)(2,5,6,8)(9,10,12,15,14,16,13,11);(2,6)(5,8)(9,11,12,10,14,15,13,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 923);