# Properties

 Label 39T2 Degree $39$ Order $39$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_{13}:C_3$

## Group action invariants

 Degree $n$: $39$ Transitive number $t$: $2$ Group: $C_{13}:C_3$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $39$ Generators: (1,14,6)(2,15,4)(3,13,5)(7,30,23)(8,28,24)(9,29,22)(10,16,31)(11,17,32)(12,18,33)(19,20,21)(25,35,37)(26,36,38)(27,34,39), (1,3,2)(4,11,30)(5,12,28)(6,10,29)(7,21,18)(8,19,16)(9,20,17)(13,37,32)(14,38,33)(15,39,31)(22,25,36)(23,26,34)(24,27,35)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 13: $C_{13}:C_3$

## Low degree siblings

13T3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 2, 3)( 4,30,11)( 5,28,12)( 6,29,10)( 7,18,21)( 8,16,19)( 9,17,20) (13,32,37)(14,33,38)(15,31,39)(22,36,25)(23,34,26)(24,35,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 3, 2)( 4,11,30)( 5,12,28)( 6,10,29)( 7,21,18)( 8,19,16)( 9,20,17) (13,37,32)(14,38,33)(15,39,31)(22,25,36)(23,26,34)(24,27,35)$ $13, 13, 13$ $3$ $13$ $( 1, 4, 7,12,13,17,19,24,25,29,31,34,38)( 2, 5, 8,10,14,18,20,22,26,30,32,35, 39)( 3, 6, 9,11,15,16,21,23,27,28,33,36,37)$ $13, 13, 13$ $3$ $13$ $( 1, 7,13,19,25,31,38, 4,12,17,24,29,34)( 2, 8,14,20,26,32,39, 5,10,18,22,30, 35)( 3, 9,15,21,27,33,37, 6,11,16,23,28,36)$ $13, 13, 13$ $3$ $13$ $( 1,13,25,38,12,24,34, 7,19,31, 4,17,29)( 2,14,26,39,10,22,35, 8,20,32, 5,18, 30)( 3,15,27,37,11,23,36, 9,21,33, 6,16,28)$ $13, 13, 13$ $3$ $13$ $( 1,24, 4,25, 7,29,12,31,13,34,17,38,19)( 2,22, 5,26, 8,30,10,32,14,35,18,39, 20)( 3,23, 6,27, 9,28,11,33,15,36,16,37,21)$

## Group invariants

 Order: $39=3 \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [39, 1]
 Character table:  3 1 1 1 . . . . 13 1 . . 1 1 1 1 1a 3a 3b 13a 13b 13c 13d 2P 1a 3b 3a 13b 13c 13d 13a 3P 1a 1a 1a 13a 13b 13c 13d 5P 1a 3b 3a 13b 13c 13d 13a 7P 1a 3a 3b 13d 13a 13b 13c 11P 1a 3b 3a 13d 13a 13b 13c 13P 1a 3a 3b 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 X.2 1 A /A 1 1 1 1 X.3 1 /A A 1 1 1 1 X.4 3 . . B C /B /C X.5 3 . . C /B /C B X.6 3 . . /B /C B C X.7 3 . . /C B C /B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(13)+E(13)^3+E(13)^9 C = E(13)^2+E(13)^5+E(13)^6