# Properties

 Label 13T2 Degree $13$ Order $26$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $D_{13}$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

26T2

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

## Group invariants

 Order: $26=2 \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [26, 1]
 Character table:  2 1 1 . . . . . . 13 1 . 1 1 1 1 1 1 1a 2a 13a 13b 13c 13d 13e 13f 2P 1a 1a 13b 13d 13f 13e 13c 13a 3P 1a 2a 13c 13f 13d 13a 13b 13e 5P 1a 2a 13e 13c 13b 13f 13a 13d 7P 1a 2a 13f 13a 13e 13b 13d 13c 11P 1a 2a 13b 13d 13f 13e 13c 13a 13P 1a 2a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 1 1 1 X.3 2 . A C B D F E X.4 2 . B E D A C F X.5 2 . C D E F B A X.6 2 . D F A B E C X.7 2 . E A F C D B X.8 2 . F B C E A D A = E(13)^3+E(13)^10 B = E(13)^4+E(13)^9 C = E(13)^6+E(13)^7 D = E(13)+E(13)^12 E = E(13)^5+E(13)^8 F = E(13)^2+E(13)^11