C₃×#509 #5383

order	192
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	Aut(G) = C₂×#61451 (order 512) · Inn(G) = C₂×D₈ · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂×C₃	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂³×C₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ G

90 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)(3,6)(4,13)(5,7)(8,10)(9,11)(12,14)(15,16)(17,18,19);(2,6)(3,10)(4,11)(7,14)(8,9)(12,16)(13,15);(2,7)(6,14)(8,15)(9,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 942);