C₂×C₃×#392 #84248

order	384
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

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

1812 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.

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

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16824);