C₃×#2057 #72772

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

Aliases

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

282 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

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

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5348);