Properties

Label 26T9
Order \(156\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times D_{13}:C_3$

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Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $9$
Group :  $C_2\times D_{13}:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,20,4,26,12)(2,5,19,3,25,11)(7,14,21,23,17,10)(8,13,22,24,18,9)(15,16), (1,13,3,7,21,5)(2,14,4,8,22,6)(9,16,24,26,20,11)(10,15,23,25,19,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
12:  $C_6\times C_2$
78:  $C_{13}:C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $C_{13}:C_6$

Low degree siblings

26T9

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

Group invariants

Order:  $156=2^{2} \cdot 3 \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [156, 8]
Character table:   
      2  2  2   2  2   2  2  2   2   2   2   2  2   1   1   1   1
      3  1  1   1  1   1  1  1   1   1   1   1  1   .   .   .   .
     13  1  .   .  .   .  .  1   .   .   .   .  .   1   1   1   1

        1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c 13a 26a 13b 26b
     2P 1a 3b  3a 3a  3b 1a 1a  3b  3a  3a  3b 1a 13b 13b 13a 13a
     3P 1a 1a  2a 1a  2a 2a 2b  2b  2c  2b  2c 2c 13a 26a 13b 26b
     5P 1a 3b  6b 3a  6a 2a 2b  6e  6f  6c  6d 2c 13b 26b 13a 26a
     7P 1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c 13b 26b 13a 26a
    11P 1a 3b  6b 3a  6a 2a 2b  6e  6f  6c  6d 2c 13b 26b 13a 26a
    13P 1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c  1a  2b  1a  2b
    17P 1a 3b  6b 3a  6a 2a 2b  6e  6f  6c  6d 2c 13a 26a 13b 26b
    19P 1a 3a  6a 3b  6b 2a 2b  6c  6d  6e  6f 2c 13b 26b 13a 26a
    23P 1a 3b  6b 3a  6a 2a 2b  6e  6f  6c  6d 2c 13a 26a 13b 26b

X.1      1  1   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  -1   1  -1
X.3      1  1  -1  1  -1 -1  1   1  -1   1  -1 -1   1   1   1   1
X.4      1  1   1  1   1  1 -1  -1  -1  -1  -1 -1   1  -1   1  -1
X.5      1  A -/A /A  -A -1 -1  -A  /A -/A   A  1   1  -1   1  -1
X.6      1 /A  -A  A -/A -1 -1 -/A   A  -A  /A  1   1  -1   1  -1
X.7      1  A -/A /A  -A -1  1   A -/A  /A  -A -1   1   1   1   1
X.8      1 /A  -A  A -/A -1  1  /A  -A   A -/A -1   1   1   1   1
X.9      1  A  /A /A   A  1 -1  -A -/A -/A  -A -1   1  -1   1  -1
X.10     1 /A   A  A  /A  1 -1 -/A  -A  -A -/A -1   1  -1   1  -1
X.11     1  A  /A /A   A  1  1   A  /A  /A   A  1   1   1   1   1
X.12     1 /A   A  A  /A  1  1  /A   A   A  /A  1   1   1   1   1
X.13     6  .   .  .   .  . -6   .   .   .   .  .   B  -B  *B -*B
X.14     6  .   .  .   .  . -6   .   .   .   .  .  *B -*B   B  -B
X.15     6  .   .  .   .  .  6   .   .   .   .  .   B   B  *B  *B
X.16     6  .   .  .   .  .  6   .   .   .   .  .  *B  *B   B   B

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(13)^2+E(13)^5+E(13)^6+E(13)^7+E(13)^8+E(13)^11
  = (-1-Sqrt(13))/2 = -1-b13