C₃×#1693 #72408

order	384
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃
prime radical	C₂×C₃×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

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

normal subgroups only

This group has 166 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) = 192

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4984);