Properties

Label 26T31
Order \(4056\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $31$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,5,24,12,20,8,26)(2,22,10,23,11,15,3,14)(4,19,7,21,9,18,6,16)(13,25), (1,15,5,14,11,19,7,20)(2,18,13,25,10,16,12,22)(3,21,8,23,9,26,4,24)(6,17)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
8:  $C_8$
12:  $C_3 : C_4$
24:  24T8

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 $ $1$ $1$ $()$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
$ 13, 13 $ $24$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $24$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $24$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $24$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $24$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $24$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $169$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $169$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $338$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 6, 6, 6, 6, 1, 1 $ $338$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,20,24,22,23,16,26,21,17,19,18,25)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,21,24,19,23,25,26,20,17,22,18,16)$
$ 8, 8, 8, 2 $ $507$ $8$ $( 1,17, 5,24,12,20, 8,26)( 2,22,10,23,11,15, 3,14)( 4,19, 7,21, 9,18, 6,16) (13,25)$
$ 8, 8, 8, 2 $ $507$ $8$ $( 1,24, 3,14,13,16,11,26)( 2,19, 8,15,12,21, 6,25)( 4,22, 5,17,10,18, 9,23) ( 7,20)$
$ 8, 8, 8, 2 $ $507$ $8$ $( 1,16, 3,14, 6,24, 4,26)( 2,15,11,19, 5,25, 9,21)( 7,23,12,18,13,17, 8,22) (10,20)$
$ 8, 8, 8, 2 $ $507$ $8$ $( 1,25, 6,17, 7,18, 2,26)( 3,14, 9,20, 5,16,12,23)( 4,15)( 8,19,10,21,13,24, 11,22)$

Group invariants

Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table:   
      2  3   .   .   .   .   .   .   .  3  3  3  2  2   2   2   3   3   3   3
      3  1   .   .   .   .   .   .   .  1  1  1  1  1   1   1   .   .   .   .
     13  2   2   2   2   2   2   2   2  .  .  .  .  .   .   .   .   .   .   .

        1a 13a 13b 13c 13d 13e 13f 13g 2a 4a 4b 3a 6a 12a 12b  8a  8b  8c  8d
     2P 1a 13a 13f 13b 13g 13d 13c 13e 1a 2a 2a 3a 3a  6a  6a  4a  4a  4b  4b
     3P 1a 13a 13f 13b 13g 13d 13c 13e 2a 4b 4a 1a 2a  4b  4a  8c  8d  8a  8b
     5P 1a 13a 13b 13c 13d 13e 13f 13g 2a 4a 4b 3a 6a 12a 12b  8b  8a  8d  8c
     7P 1a 13a 13c 13f 13e 13g 13b 13d 2a 4b 4a 3a 6a 12b 12a  8d  8c  8b  8a
    11P 1a 13a 13f 13b 13g 13d 13c 13e 2a 4b 4a 3a 6a 12b 12a  8c  8d  8a  8b
    13P 1a  1a  1a  1a  1a  1a  1a  1a 2a 4a 4b 3a 6a 12a 12b  8b  8a  8d  8c

X.1      1   1   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  -1  -1  -1
X.3      1   1   1   1   1   1   1   1 -1  D -D  1 -1   D  -D   F  -F -/F  /F
X.4      1   1   1   1   1   1   1   1 -1  D -D  1 -1   D  -D  -F   F  /F -/F
X.5      1   1   1   1   1   1   1   1 -1 -D  D  1 -1  -D   D -/F  /F   F  -F
X.6      1   1   1   1   1   1   1   1 -1 -D  D  1 -1  -D   D  /F -/F  -F   F
X.7      1   1   1   1   1   1   1   1  1 -1 -1  1  1  -1  -1   D   D  -D  -D
X.8      1   1   1   1   1   1   1   1  1 -1 -1  1  1  -1  -1  -D  -D   D   D
X.9      2   2   2   2   2   2   2   2  2 -2 -2 -1 -1   1   1   .   .   .   .
X.10     2   2   2   2   2   2   2   2  2  2  2 -1 -1  -1  -1   .   .   .   .
X.11     2   2   2   2   2   2   2   2 -2  E -E -1  1  -D   D   .   .   .   .
X.12     2   2   2   2   2   2   2   2 -2 -E  E -1  1   D  -D   .   .   .   .
X.13    24  11  -2  -2  -2  -2  -2  -2  .  .  .  .  .   .   .   .   .   .   .
X.14    24  -2   A   B  -2  -2   C  -2  .  .  .  .  .   .   .   .   .   .   .
X.15    24  -2   B   C  -2  -2   A  -2  .  .  .  .  .   .   .   .   .   .   .
X.16    24  -2   C   A  -2  -2   B  -2  .  .  .  .  .   .   .   .   .   .   .
X.17    24  -2  -2  -2   A   B  -2   C  .  .  .  .  .   .   .   .   .   .   .
X.18    24  -2  -2  -2   B   C  -2   A  .  .  .  .  .   .   .   .   .   .   .
X.19    24  -2  -2  -2   C   A  -2   B  .  .  .  .  .   .   .   .   .   .   .

A = -4*E(13)^2-4*E(13)^3-3*E(13)^4-3*E(13)^6-3*E(13)^7-3*E(13)^9-4*E(13)^10-4*E(13)^11
B = -4*E(13)-3*E(13)^2-3*E(13)^3-4*E(13)^5-4*E(13)^8-3*E(13)^10-3*E(13)^11-4*E(13)^12
C = -3*E(13)-4*E(13)^4-3*E(13)^5-4*E(13)^6-4*E(13)^7-3*E(13)^8-4*E(13)^9-3*E(13)^12
D = -E(4)
  = -Sqrt(-1) = -i
E = -2*E(4)
  = -2*Sqrt(-1) = -2i
F = -E(8)^3