C₁₃×#13618 #15699957

order	3328
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#13618

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₃
prime radical	C₂³×C₁₃	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂²×C₁₃×C₁₆ ◁ G

290 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 126 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)(20,31)(21,26)(22,40)(23,44)(24,30)(25,33)(27,42)(28,38)(29,39)(32,59)(34,65)(35,61)(36,60)(37,70)(41,56)(43,58)(45,53)(46,52)(47,62)(48,75)(49,68)(50,64)(51,72)(54,81)(55,63)(57,66)(67,79)(69,80)(71,78)(73,74)(76,77);(18,20,24,35)(19,25,30,50)(21,36,39,32)(22,41,43,76)(23,45,46,69)(26,49,29,47)(27,53,55,80)(28,56,57,77)(31,58,61,40)(33,63,64,42)(34,66,67,38)(37,71,73,75)(44,48,52,78)(51,65,54,79)(59,70,60,74)(62,72,68,81);(18,21,24,39)(19,26,30,29)(20,32,35,36)(22,37,43,73)(23,38,46,66)(25,47,50,49)(27,51,55,54)(28,52,57,44)(31,59,61,60)(33,62,64,68)(34,45,67,69)(40,70,58,74)(41,75,76,71)(42,72,63,81)(48,77,78,56)(53,79,80,65)

Conjugacy classes & character table

676 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6606);