Properties

Label 26T49
Degree $26$
Order $11232$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no

Learn more about

Group action invariants

Degree $n$:  $26$
Transitive number $t$:  $49$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,22,8,25,5,23)(2,26,13,21,4,16)(3,24,9,17,11,19)(6,18)(7,20,10,14,12,15), (1,24,12,23,3,17,10,25,13,21,5,18)(2,20)(4,14,8,16,11,26)(6,15,7,19)(9,22)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $117$ $2$ $( 1, 5)( 2, 7)( 3,12)(10,13)(14,24)(15,19)(16,21)(17,26)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ $702$ $4$ $( 1,13, 5,10)( 2, 3, 7,12)( 4, 9)( 8,11)(14,17,24,26)(15,21,19,16)(18,25) (22,23)$
$ 8, 8, 4, 4, 2 $ $1404$ $8$ $( 1,15,13,21, 5,19,10,16)( 2,24, 3,26, 7,14,12,17)( 4,18, 9,25)( 6,20) ( 8,22,11,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $234$ $2$ $( 1,18)( 2,19)( 3,17)( 4,21)( 5,16)( 6,24)( 7,23)( 8,15)( 9,26)(10,14)(11,25) (12,22)(13,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $624$ $3$ $( 1,11, 4)( 2,12, 9)( 5, 6,13)( 7, 8,10)(14,23,15)(16,24,20)(18,25,21) (19,22,26)$
$ 6, 6, 6, 6, 2 $ $1872$ $6$ $( 1,21,11,18, 4,25)( 2,26,12,19, 9,22)( 3,17)( 5,20, 6,16,13,24) ( 7,14, 8,23,10,15)$
$ 13, 13 $ $864$ $13$ $( 1, 5, 9, 8,12,11, 2, 7, 3, 6,10, 4,13)(14,22,17,23,25,16,21,15,18,19,20,26, 24)$
$ 13, 13 $ $864$ $13$ $( 1, 3, 8, 4, 2, 5, 6,12,13, 7, 9,10,11)(14,18,23,26,21,22,19,25,24,15,17,20, 16)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $104$ $3$ $( 1, 2,10)( 3, 6,12)( 5,13, 7)(14,25,24)(15,22,19)(17,18,26)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2 $ $234$ $4$ $( 1,15)( 2,22)( 3,18,13,24)( 4,20)( 5,25,12,17)( 6,26, 7,14)( 8,21,11,23) ( 9,16)(10,19)$
$ 6, 6, 3, 3, 2, 2, 1, 1, 1, 1 $ $936$ $6$ $( 1,10, 2)( 3, 5, 6,13,12, 7)( 8,11)(14,18,25,26,24,17)(15,19,22)(21,23)$
$ 12, 6, 4, 2, 2 $ $936$ $12$ $( 1,22,10,15, 2,19)( 3,14, 5,18, 6,25,13,26,12,24, 7,17)( 4,20)( 8,23,11,21) ( 9,16)$
$ 12, 6, 4, 2, 2 $ $936$ $12$ $( 1,22,10,15, 2,19)( 3,26, 5,24, 6,17,13,14,12,18, 7,25)( 4,20)( 8,21,11,23) ( 9,16)$
$ 8, 8, 4, 4, 1, 1 $ $1404$ $8$ $( 2,12, 6,13, 4,11, 5, 9)( 3,10, 7, 8)(14,19,22,26,23,25,21,20)(15,16,24,18)$

Group invariants

Order:  $11232=2^{5} \cdot 3^{3} \cdot 13$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table:   
      2  5  4  1  1  2  5  4  3  4  2   2   2   .   .  3
      3  3  1  2  1  3  1  .  .  1  1   1   1   .   .  .
     13  1  .  .  .  .  .  .  .  .  .   .   .   1   1  .

        1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12a 12b 13a 13b 8b
     2P 1a 1a 3a 3a 3b 1a 2b 4a 2b 3b  6b  6b 13b 13a 4a
     3P 1a 2a 1a 2a 1a 2b 4a 8a 4b 2b  4b  4b 13a 13b 8b
     5P 1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12b 12a 13b 13a 8b
     7P 1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12b 12a 13b 13a 8b
    11P 1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12a 12b 13b 13a 8b
    13P 1a 2a 3a 6a 3b 2b 4a 8a 4b 6b 12a 12b  1a  1a 8b

X.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
X.3     12  .  .  .  3  4  .  .  .  1   A  -A  -1  -1  .
X.4     12  .  .  .  3  4  .  .  .  1  -A   A  -1  -1  .
X.5     13 -1  1 -1  4 -3  1  1  3  .   .   .   .   . -1
X.6     13  1  1  1  4 -3  1 -1 -3  .   .   .   .   . -1
X.7     26 -2 -1  1 -1  2  2  .  2 -1  -1  -1   .   .  .
X.8     26  2 -1 -1 -1  2  2  . -2 -1   1   1   .   .  .
X.9     27 -3  .  .  .  3 -1  1 -3  .   .   .   1   1 -1
X.10    27  3  .  .  .  3 -1 -1  3  .   .   .   1   1 -1
X.11    32  .  2  . -4  .  .  .  .  .   .   .   B  *B  .
X.12    32  .  2  . -4  .  .  .  .  .   .   .  *B   B  .
X.13    39 -3  .  .  3 -1 -1 -1  1 -1   1   1   .   .  1
X.14    39  3  .  .  3 -1 -1  1 -1 -1  -1  -1   .   .  1
X.15    52  . -2  . -2 -4  .  .  .  2   .   .   .   .  .

A = -E(12)^7+E(12)^11
  = Sqrt(3) = r3
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