C₃×#2213 #72928

order	384
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂×C₃	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ G

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

normal subgroups only

This group has 170 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5504);