Properties

 Label 13T3 Degree $13$ Order $39$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{13}:C_3$

Related objects

Group action invariants

 Degree $n$: $13$ Transitive number $t$: $3$ Group: $C_{13}:C_3$ CHM label: $F_{39}(13)=13:3$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $\card{\Aut(F/K)}$: $1$ Generators: (1,2,3,4,5,6,7,8,9,10,11,12,13), (1,3,9)(2,6,5)(4,12,10)(7,8,11)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

39T2

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 1$ $13$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)$ $3, 3, 3, 3, 1$ $13$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)$ $13$ $3$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$ $13$ $3$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$ $13$ $3$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$ $13$ $3$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)$

Group invariants

 Order: $39=3 \cdot 13$ Cyclic: no Abelian: no Solvable: yes Label: 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