C₁₃×#13666 #15700005

order	3328
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#13666

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃
prime radical	C₂×C₁₃×#508	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂²×C₁₃×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) = 81
transitive degree	μ_t(G) = 1664

A minimal faithful permutation representation — 3 generators on 81 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,17)(18,19,24,30)(20,31,36,62)(21,26,39,29)(22,40,43,74)(23,44,46,52)(25,47,50,33)(27,53,56,42)(28,57,59,38)(32,61,35,68)(34,66,67,80)(37,69,72,73)(41,58,76,81)(45,54,64,79)(48,65,60,63)(49,70,77,75)(51,78,55,71);(18,20)(19,25)(21,32)(22,41)(23,45)(24,36)(26,48)(27,54)(28,58)(29,60)(30,50)(31,40)(33,42)(34,38)(35,39)(37,70)(43,76)(44,49)(46,64)(47,53)(51,66)(52,77)(55,80)(56,79)(57,67)(59,81)(61,69)(62,74)(63,71)(65,78)(68,73)(72,75);(15,17)(18,21,24,39)(19,26,30,29)(20,32,36,35)(22,37,43,72)(23,38,46,57)(25,48,50,60)(27,51,56,55)(28,52,59,44)(31,61,62,68)(33,63,47,65)(34,64,67,45)(40,69,74,73)(41,70,76,75)(42,71,53,78)(49,58,77,81)(54,66,79,80)

Conjugacy classes & character table

637 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6654);