C₃×#10236 #10628843

order	768
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂²×C₃×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

628 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3224);