C₃×#13666 #10632273

order	768
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13666

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₃
prime radical	C₂×C₃×#508	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂²×C₃×C₁₆ ◁ G

288 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 78 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)(8,9,14,20)(10,21,26,52)(11,16,29,19)(12,30,33,64)(13,34,36,42)(15,37,40,23)(17,43,46,32)(18,47,49,28)(22,51,25,58)(24,56,57,70)(27,59,62,63)(31,48,66,71)(35,44,54,69)(38,55,50,53)(39,60,67,65)(41,68,45,61);(8,10)(9,15)(11,22)(12,31)(13,35)(14,26)(16,38)(17,44)(18,48)(19,50)(20,40)(21,30)(23,32)(24,28)(25,29)(27,60)(33,66)(34,39)(36,54)(37,43)(41,56)(42,67)(45,70)(46,69)(47,57)(49,71)(51,59)(52,64)(53,61)(55,68)(58,63)(62,65);(5,7)(8,11,14,29)(9,16,20,19)(10,22,26,25)(12,27,33,62)(13,28,36,47)(15,38,40,50)(17,41,46,45)(18,42,49,34)(21,51,52,58)(23,53,37,55)(24,54,57,35)(30,59,64,63)(31,60,66,65)(32,61,43,68)(39,48,67,71)(44,56,69,70)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6654);