C₁₃×#13615 #15699954

order	3328
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#13615

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	C₂×C₁₃×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ C₂²×C₁₃×C₁₆ ◁ G

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

normal subgroups only

This group has 298 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)(32,58)(34,64)(35,66)(36,59)(37,67)(41,55)(43,73)(45,52)(46,51)(47,60)(48,74)(49,63)(50,69)(53,79)(54,76)(56,71)(57,62)(61,78)(65,80)(68,77)(70,72)(75,81);(18,20)(19,25)(21,32)(22,41)(23,45)(24,36)(26,47)(27,52)(28,55)(29,57)(30,49)(31,40)(33,42)(34,38)(35,39)(37,68)(43,75)(44,48)(46,61)(50,64)(51,77)(53,80)(54,78)(56,81)(58,67)(59,73)(60,69)(62,79)(63,76)(65,71)(66,72)(70,74);(18,21,24,39)(19,26,30,29)(20,32,36,35)(22,37,43,70)(23,38,46,71)(25,47,49,57)(27,50,54,53)(28,51,56,44)(31,58,59,66)(33,60,63,62)(34,61,65,45)(40,67,73,72)(41,68,75,74)(42,69,76,79)(48,55,77,81)(52,64,78,80)

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, 6603);