C₃×#2081 #72796

order	384
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂×C₃	#397
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ G

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

normal subgroups only

This group has 204 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) = 48

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5372);