C₃×#2566 #84430

order	384
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

548 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17006);