C₃×#2054 #72769

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	C₂²×C₃×D₈	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

360 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 104 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)(3,4)(10,15)(11,16)(12,13)(17,18,19);(5,6)(7,8)(9,10)(11,14)(12,16)(13,15);(2,4)(6,8)(9,11,12,16,14,10,13,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5345);