C₃×#1626 #72341

order	384
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃
prime radical	C₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

210 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4917);