C₃×#557 #5887

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 6144
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

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

200 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1446);