C₃×#2050 #72765

order	384
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center	C₃×C₄	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂×C₃	#394
prime radical	C₃×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

266 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5341);