C₃×#2002 #72717

order	384
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ G

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

normal subgroups only

This group has 478 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) = 15
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5293);