C₁₃×#13633 #15699972

order	3328
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃×#13633

Subgroups

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

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

normal subgroups only

This group has 190 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)(21,27)(22,40)(23,44)(24,46)(25,30)(26,34)(28,32)(29,41)(33,53)(35,66)(36,68)(37,56)(38,69)(39,52)(42,72)(43,71)(45,73)(47,61)(48,75)(49,65)(50,58)(51,60)(54,57)(55,63)(59,70)(62,78)(64,74)(67,81)(76,79)(77,80);(18,20,32,57,23,35,38,61,21,33,58,79,39,62,43,65,25,37,60,80,45,67,42,64,24,36,59,75,41,63,22,34)(19,26,40,55,29,48,70,68,46,74,72,81,73,77,51,56,30,49,71,78,52,76,50,53,27,47,69,66,44,54,28,31);(18,21,25,24)(19,27,30,46)(20,33,37,36)(22,38,43,42)(23,39,45,41)(26,47,49,74)(28,50,51,70)(29,44,52,73)(31,53,56,68)(32,58,60,59)(34,61,65,64)(35,62,67,63)(40,69,71,72)(48,54,76,77)(55,66,78,81)(57,79,80,75)

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 6621);