C₂×C₃×#393 #84249

order	384
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

1036 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 328 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) = 192

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16825);