C₃×#2031 #72746

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	#393
prime radical	C₂×C₃×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

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

normal subgroups only

This group has 244 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)(3,10,12,16)(4,8)(5,14)(6,9)(7,13,15,11)(17,18,19);(3,11,12,13)(7,10,15,16);(1,3)(2,7)(4,11)(5,13)(6,12)(8,16)(9,15)(10,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5322);