Properties

Label 13T6
Order \(156\)
n \(13\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $F_{13}$

Related objects

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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)
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)
$|\Aut(F/K)|$:  $1$

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 TypeSizeOrderRepresentative
$ 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