C₃×#529 #5859

order	192
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = C₂⁵×#120 (order 1024) · Inn(G) = C₂³ · Out(G) = C₂⁷
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center	C₂×C₃×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
socle	C₂²×C₃	C₂⁴
prime radical	C₃×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

234 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1418);