C₃×#542 #5872

order	192
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	Aut(G) = C₂⁵×#120 (order 1024) · Inn(G) = C₂⁴ · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

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

216 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)(10,13)(11,15)(17,18,19);(1,3,4,7)(2,5,6,8)(9,10,12,13)(11,14,15,16);(2,6)(5,8)(10,13)(14,16);(9,11,12,15)(10,14,13,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1431);