C₃×#2062 #72777

order	384
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#120
commutator, Frattini, cosocle	C₂⁴	C₂³×C₃
prime radical	C₂⁴×C₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ G

314 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5353);