C₃×#144386 #72255271

order	1536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#2021
commutator, Frattini, cosocle	C₂³×D₄	C₂³×C₃
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×D₄ ◁ G

3576 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:

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

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,4,6)(3,5)(7,8)(10,13)(17,18,19), (5,8)(10,14)(13,16), (1,3,4,7)(2,5)(6,8)(9,10,15,16)(11,14,12,13));