C₃×#2102 #72817

order	384
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center	C₃×C₄	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂×C₃	#400
prime radical	C₂³×C₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

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

normal subgroups only

This group has 158 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5393);