C₂×C₃×#500 #85023

order	384
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

424 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17599);