C₁₃×#13315 #15699654

order	3328
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₃×#13315

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂×C₁₃×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#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) = 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)(18,19,25,31)(20,32,37,63)(21,38,42,60)(22,43,47,58)(23,30,24,29)(26,49,53,34)(27,48,56,41)(28,45,59,46)(33,64,66,81)(35,57,36,62)(39,61,68,79)(40,71,74,55)(44,51,67,52)(50,76,77,70)(54,72,78,65)(69,75,80,73);(18,20,25,37)(19,26,31,53)(21,39,42,68)(22,44,47,67)(23,36,24,35)(27,54,56,78)(28,57,59,62)(29,52,30,51)(32,58,63,43)(33,48,66,41)(34,45,49,46)(38,50,60,77)(40,72,74,65)(55,79,71,61)(64,80,81,69)(70,75,76,73);(15,17)(18,21)(19,27)(20,33)(22,40)(23,48)(24,41)(25,42)(26,50)(28,55)(29,60)(30,38)(31,56)(32,61)(34,65)(35,68)(36,39)(37,66)(43,69)(44,70)(45,75)(46,73)(47,74)(49,72)(51,78)(52,54)(53,77)(57,64)(58,80)(59,71)(62,81)(63,79)(67,76)

Conjugacy classes & character table

520 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6303);