C₃×#2208 #72923

order	384
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃
prime radical	C₂×C₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈ ◁ G

166 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(2,4)(5,6,9,13)(7,10,16,19)(8,17,15,12)(11,20,18,14)(21,22,23);(1,2,3,4);(6,10)(7,14)(8,15)(11,12)(13,19)(16,20)(17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5499);