C₃×#2250 #72965

order	384
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

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

198 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 72 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)(6,10)(7,11)(8,14)(12,16)(13,20)(15,19)(17,18)(21,22,23);(5,6,11,18,8,13,15,16,9,12,19,20,14,17,7,10);(1,2,3,4)(6,12)(10,16)(13,17)(18,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5541);