C₃×#2501 #84365

order	384
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂×C₃×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

708 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.

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

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16941);