C₃×#13315 #10631922

order	768
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13315

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂×C₃×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#131 ◁ G

516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2,3)(4,5)(6,7)(8,9,15,21)(10,22,27,53)(11,28,32,50)(12,33,37,48)(13,20,14,19)(16,39,43,24)(17,38,46,31)(18,35,49,36)(23,54,56,71)(25,47,26,52)(29,51,58,69)(30,61,64,45)(34,41,57,42)(40,66,67,60)(44,62,68,55)(59,65,70,63);(8,10,15,27)(9,16,21,43)(11,29,32,58)(12,34,37,57)(13,26,14,25)(17,44,46,68)(18,47,49,52)(19,42,20,41)(22,48,53,33)(23,38,56,31)(24,35,39,36)(28,40,50,67)(30,62,64,55)(45,69,61,51)(54,70,71,59)(60,65,66,63);(5,7)(8,11)(9,17)(10,23)(12,30)(13,38)(14,31)(15,32)(16,40)(18,45)(19,50)(20,28)(21,46)(22,51)(24,55)(25,58)(26,29)(27,56)(33,59)(34,60)(35,65)(36,63)(37,64)(39,62)(41,68)(42,44)(43,67)(47,54)(48,70)(49,61)(52,71)(53,69)(57,66)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6303);