# Properties

 Label 13T6 Degree $13$ Order $156$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $F_{13}$

# Related objects

## Group action invariants

 Degree $n$: $13$ Transitive number $t$: $6$ Group: $F_{13}$ CHM label: $F_{156}(13)=13:12$ 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), (1,2,4,8,3,6,12,11,9,5,10,7)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

26T8, 39T11

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

## Group invariants

 Order: $156=2^{2} \cdot 3 \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [156, 7]
 Character table:  2 2 2 2 2 2 2 2 2 2 2 2 2 . 3 1 1 1 1 1 1 1 1 1 1 1 1 . 13 1 . . . . . . . . . . . 1 1a 12a 3a 6a 4a 12b 12c 4b 3b 6b 12d 2a 13a 2P 1a 6a 3b 3a 2a 6b 6b 2a 3a 3b 6a 1a 13a 3P 1a 4b 1a 2a 4b 4b 4a 4a 1a 2a 4a 2a 13a 5P 1a 12b 3b 6b 4a 12a 12d 4b 3a 6a 12c 2a 13a 7P 1a 12d 3a 6a 4b 12c 12b 4a 3b 6b 12a 2a 13a 11P 1a 12c 3b 6b 4b 12d 12a 4a 3a 6a 12b 2a 13a 13P 1a 12a 3a 6a 4a 12b 12c 4b 3b 6b 12d 2a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 1 X.3 1 A -A -/A -1 /A /A -1 -/A -A A 1 1 X.4 1 /A -/A -A -1 A A -1 -A -/A /A 1 1 X.5 1 -/A -/A -A 1 -A -A 1 -A -/A -/A 1 1 X.6 1 -A -A -/A 1 -/A -/A 1 -/A -A -A 1 1 X.7 1 B 1 -1 B B -B -B 1 -1 -B -1 1 X.8 1 -B 1 -1 -B -B B B 1 -1 B -1 1 X.9 1 C -A /A B -/C /C -B -/A A -C -1 1 X.10 1 -/C -/A A B C -C -B -A /A /C -1 1 X.11 1 /C -/A A -B -C C B -A /A -/C -1 1 X.12 1 -C -A /A -B /C -/C B -/A A C -1 1 X.13 12 . . . . . . . . . . . -1 A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = -E(4) = -Sqrt(-1) = -i C = -E(12)^7