C₃×#1597 #72312

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×#447	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ G

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

normal subgroups only

This group has 272 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) = 96

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4888);