C₃×#1697 #72412

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	C₃×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

170 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,5)(4,6)(7,8)(10,13)(11,15)(14,16)(17,18,19);(2,5)(6,7)(9,10,11,13,12,14,15,16);(2,6)(4,8)(5,7)(10,14)(13,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4988);