Properties

Label 26T8
Order \(156\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_{13}$

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Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $8$
Group :  $F_{13}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,6,7,9,12,14,16,18,19,21,24,25)(2,4,5,8,10,11,13,15,17,20,22,23,26), (1,4,7,15,6,11,24,22,18,10,19,13)(2,3,8,16,5,12,23,21,17,9,20,14)(25,26)
$|\Aut(F/K)|$:  $2$

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

Degree 2: $C_2$

Degree 13: $F_{13}$

Low degree siblings

13T6, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $13$ $3$ $( 3, 7,19)( 4, 8,20)( 5,13,11)( 6,14,12)( 9,25,21)(10,26,22)(15,17,23) (16,18,24)$
$ 6, 6, 6, 6, 1, 1 $ $13$ $6$ $( 3, 9, 7,25,19,21)( 4,10, 8,26,20,22)( 5,17,13,23,11,15)( 6,18,14,24,12,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $13$ $3$ $( 3,19, 7)( 4,20, 8)( 5,11,13)( 6,12,14)( 9,21,25)(10,22,26)(15,23,17) (16,24,18)$
$ 6, 6, 6, 6, 1, 1 $ $13$ $6$ $( 3,21,19,25, 7, 9)( 4,22,20,26, 8,10)( 5,15,11,23,13,17)( 6,16,12,24,14,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $13$ $2$ $( 3,25)( 4,26)( 5,23)( 6,24)( 7,21)( 8,22)( 9,19)(10,20)(11,17)(12,18)(13,15) (14,16)$
$ 12, 12, 2 $ $13$ $12$ $( 1, 2)( 3, 5, 9,17, 7,13,25,23,19,11,21,15)( 4, 6,10,18, 8,14,26,24,20,12,22, 16)$
$ 4, 4, 4, 4, 4, 4, 2 $ $13$ $4$ $( 1, 2)( 3,11,25,17)( 4,12,26,18)( 5,21,23, 7)( 6,22,24, 8)( 9,15,19,13) (10,16,20,14)$
$ 12, 12, 2 $ $13$ $12$ $( 1, 2)( 3,13,21,17,19, 5,25,15, 7,11, 9,23)( 4,14,22,18,20, 6,26,16, 8,12,10, 24)$
$ 12, 12, 2 $ $13$ $12$ $( 1, 2)( 3,15,21,11,19,23,25,13, 7,17, 9, 5)( 4,16,22,12,20,24,26,14, 8,18,10, 6)$
$ 4, 4, 4, 4, 4, 4, 2 $ $13$ $4$ $( 1, 2)( 3,17,25,11)( 4,18,26,12)( 5, 7,23,21)( 6, 8,24,22)( 9,13,19,15) (10,14,20,16)$
$ 12, 12, 2 $ $13$ $12$ $( 1, 2)( 3,23, 9,11, 7,15,25, 5,19,17,21,13)( 4,24,10,12, 8,16,26, 6,20,18,22, 14)$
$ 13, 13 $ $12$ $13$ $( 1, 3, 6, 7, 9,12,14,16,18,19,21,24,25)( 2, 4, 5, 8,10,11,13,15,17,20,22,23, 26)$

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 3a  6a 3b  6b 2a 12a 4a 12b 12c 4b 12d 13a
     2P 1a 3b  3a 3a  3b 1a  6a 2a  6b  6b 2a  6a 13a
     3P 1a 1a  2a 1a  2a 2a  4b 4b  4b  4a 4a  4a 13a
     5P 1a 3b  6b 3a  6a 2a 12b 4a 12a 12d 4b 12c 13a
     7P 1a 3a  6a 3b  6b 2a 12d 4b 12c 12b 4a 12a 13a
    11P 1a 3b  6b 3a  6a 2a 12c 4b 12d 12a 4a 12b 13a
    13P 1a 3a  6a 3b  6b 2a 12a 4a 12b 12c 4b 12d  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  1  -1  1  -1 -1   B  B   B  -B -B  -B   1
X.4      1  1  -1  1  -1 -1  -B -B  -B   B  B   B   1
X.5      1  A -/A /A  -A -1   C  B -/C  /C -B  -C   1
X.6      1  A -/A /A  -A -1  -C -B  /C -/C  B   C   1
X.7      1 /A  -A  A -/A -1 -/C  B   C  -C -B  /C   1
X.8      1 /A  -A  A -/A -1  /C -B  -C   C  B -/C   1
X.9      1  A  /A /A   A  1  -A -1 -/A -/A -1  -A   1
X.10     1 /A   A  A  /A  1 -/A -1  -A  -A -1 -/A   1
X.11     1  A  /A /A   A  1   A  1  /A  /A  1   A   1
X.12     1 /A   A  A  /A  1  /A  1   A   A  1  /A   1
X.13    12  .   .  .   .  .   .  .   .   .  .   .  -1

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(4)
  = -Sqrt(-1) = -i
C = -E(12)^11