C₃×#2252 #72967

order	384
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈ ◁ G

128 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,4,6)(3,7,8,5)(10,14)(11,15)(12,18)(16,24)(17,22)(19,23)(20,21)(25,26,27);(1,3,4,8)(2,5,6,7);(9,10,15,21,12,16,19,22,13,17,23,24,18,20,11,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5543);