C₃×#2396 #84260

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

Aliases

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₂⁴×C₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

528 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,17)(19,20)(21,22,23);(1,2)(3,4)(14,17)(18,20);(2,4)(5,7,8,12)(6,9,10,11);(13,15,16,19)(14,18,17,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16836);