C₃×#24658 #10643265

order	768
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄² ◁ G

1100 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 384

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 17646);