C₃×#1613 #72328

order	384
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
prime radical	C₃×#534	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ G

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 96

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4904);