# Properties

 Label 13T1 Degree $13$ Order $13$ Cyclic yes Abelian yes Solvable yes Primitive yes $p$-group yes Group: $C_{13}$

# Related objects

## Group action invariants

 Degree $n$: $13$ Transitive number $t$: $1$ Group: $C_{13}$ CHM label: $C(13)=13$ Parity: $1$ Primitive: yes Nilpotency class: $1$ $|\Aut(F/K)|$: $13$ Generators: (1,2,3,4,5,6,7,8,9,10,11,12,13)

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $13$ $1$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$ $13$ $1$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$ $13$ $1$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)$ $13$ $1$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$ $13$ $1$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)$ $13$ $1$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)$ $13$ $1$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)$ $13$ $1$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$ $13$ $1$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)$ $13$ $1$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)$ $13$ $1$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)$ $13$ $1$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

## Group invariants

 Order: $13$ (is prime) Cyclic: yes Abelian: yes Solvable: yes GAP id: [13, 1]
 Character table:  13 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 13a 13b 13c 13d 13e 13f 13g 13h 13i 13j 13k 13l X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A B C D E F /F /E /D /C /B /A X.3 1 B D F /E /C /A A C E /F /D /B X.4 1 C F /D /A B E /E /B A D /F /C X.5 1 D /E /A C /F /B B F /C A E /D X.6 1 E /C B /F /A D /D A F /B C /E X.7 1 F /A E /B D /C C /D B /E A /F X.8 1 /F A /E B /D C /C D /B E /A F X.9 1 /E C /B F A /D D /A /F B /C E X.10 1 /D E A /C F B /B /F C /A /E D X.11 1 /C /F D A /B /E E B /A /D F C X.12 1 /B /D /F E C A /A /C /E F D B X.13 1 /A /B /C /D /E /F F E D C B A A = E(13) B = E(13)^2 C = E(13)^3 D = E(13)^4 E = E(13)^5 F = E(13)^6