C₃×#2230 #72945

order	384
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, socle	C₂×C₃	#476
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃
prime radical	C₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ G

126 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5521);