C₂×C₃×#2397 #10653151

order	768
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

	subgroup	G/H
center	C₂³×C₃×C₄	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
socle	C₂⁴×C₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

5150 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 5 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 27532);