C₃×#2001 #72716

order	384
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
prime radical	C₂⁴×C₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ G

392 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2)(3,4)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18)(25,26,27);(5,6)(7,8)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(2,4)(6,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5292);