Properties

Label 39T30
Order \(1053\)
n \(39\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $39$
Transitive number $t$ :  $30$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,39)(2,10,37)(3,11,38)(4,21,27)(5,19,25)(6,20,26)(7,29,15)(8,30,13)(9,28,14)(16,17,18)(22,36,33)(23,34,31)(24,35,32), (1,27,12,34,19,5,29,14,39,23,9,33,16)(2,25,10,35,20,6,30,15,37,24,7,31,17)(3,26,11,36,21,4,28,13,38,22,8,32,18)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
39:  $C_{13}:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 13: $C_{13}:C_3$

Low degree siblings

27T292, 39T29, 39T30

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $13$ $3$ $( 1, 3, 2)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(22,23,24) (31,32,33)(34,36,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $13$ $3$ $( 1, 2, 3)( 7, 9, 8)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,24,23) (31,33,32)(34,35,36)$
$ 13, 13, 13 $ $81$ $13$ $( 1,24, 5,26, 8,29,11,33,14,34,17,37,21)( 2,22, 6,27, 9,30,12,31,15,35,18,38, 19)( 3,23, 4,25, 7,28,10,32,13,36,16,39,20)$
$ 13, 13, 13 $ $81$ $13$ $( 1, 5, 8,11,14,17,21,24,26,29,33,34,37)( 2, 6, 9,12,15,18,19,22,27,30,31,35, 38)( 3, 4, 7,10,13,16,20,23,25,28,32,36,39)$
$ 13, 13, 13 $ $81$ $13$ $( 1, 8,14,21,26,33,37, 5,11,17,24,29,34)( 2, 9,15,19,27,31,38, 6,12,18,22,30, 35)( 3, 7,13,20,25,32,39, 4,10,16,23,28,36)$
$ 13, 13, 13 $ $81$ $13$ $( 1,14,26,37,11,24,34, 8,21,33, 5,17,29)( 2,15,27,38,12,22,35, 9,19,31, 6,18, 30)( 3,13,25,39,10,23,36, 7,20,32, 4,16,28)$
$ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ $117$ $9$ $( 4,12,28, 5,10,29, 6,11,30)( 7,19,18, 8,20,16, 9,21,17)(13,37,32,15,39,31,14, 38,33)(22,25,36)(23,26,34)(24,27,35)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1, 3, 2)( 4,10,29, 6,12,28, 5,11,30)( 7,20,18, 9,19,17, 8,21,16) (13,37,33,14,38,31,15,39,32)(22,25,35)(23,26,36)(24,27,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $117$ $3$ $( 1, 2, 3)( 4,11,30)( 5,12,28)( 6,10,29)( 7,21,18)( 8,19,16)( 9,20,17) (13,37,31)(14,38,32)(15,39,33)(22,25,34)(23,26,35)(24,27,36)$
$ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ $117$ $9$ $( 4,28,10, 6,30,12, 5,29,11)( 7,18,20, 9,17,19, 8,16,21)(13,32,39,14,33,37,15, 31,38)(22,36,25)(23,34,26)(24,35,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $117$ $3$ $( 1, 3, 2)( 4,28,11)( 5,29,12)( 6,30,10)( 7,17,20)( 8,18,21)( 9,16,19) (13,33,37)(14,31,38)(15,32,39)(22,35,27)(23,36,25)(24,34,26)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1, 2, 3)( 4,28,12, 5,29,10, 6,30,11)( 7,16,20, 8,17,21, 9,18,19) (13,31,38,15,33,37,14,32,39)(22,34,26)(23,35,27)(24,36,25)$

Group invariants

Order:  $1053=3^{4} \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1053, 51]
Character table:   
      3  4  4  4   .   .   .   .  2  2  2  2  2  2
     13  1  .  .   1   1   1   1  .  .  .  .  .  .

        1a 3a 3b 13a 13b 13c 13d 9a 9b 3c 9c 3d 9d
     2P 1a 3b 3a 13b 13c 13d 13a 9c 9d 3d 9a 3c 9b
     3P 1a 1a 1a 13a 13b 13c 13d 3a 3b 1a 3b 1a 3a
     5P 1a 3b 3a 13b 13c 13d 13a 9c 9d 3d 9a 3c 9b
     7P 1a 3a 3b 13d 13a 13b 13c 9a 9b 3c 9c 3d 9d
    11P 1a 3b 3a 13d 13a 13b 13c 9c 9d 3d 9a 3c 9b
    13P 1a 3a 3b  1a  1a  1a  1a 9a 9b 3c 9c 3d 9d

X.1      1  1  1   1   1   1   1  1  1  1  1  1  1
X.2      1  1  1   1   1   1   1  D  D  D /D /D /D
X.3      1  1  1   1   1   1   1 /D /D /D  D  D  D
X.4      3  3  3   B   C  /B  /C  .  .  .  .  .  .
X.5      3  3  3  /B  /C   B   C  .  .  .  .  .  .
X.6      3  3  3   C  /B  /C   B  .  .  .  .  .  .
X.7      3  3  3  /C   B   C  /B  .  .  .  .  .  .
X.8     13  A /A   .   .   .   . /D  1  D  D /D  1
X.9     13 /A  A   .   .   .   .  D  1 /D /D  D  1
X.10    13  A /A   .   .   .   .  1  D /D  1  D /D
X.11    13 /A  A   .   .   .   .  1 /D  D  1 /D  D
X.12    13  A /A   .   .   .   .  D /D  1 /D  1  D
X.13    13 /A  A   .   .   .   . /D  D  1  D  1 /D

A = 2*E(3)-E(3)^2
  = (-1+3*Sqrt(-3))/2 = 1+3b3
B = E(13)^2+E(13)^5+E(13)^6
C = E(13)^4+E(13)^10+E(13)^12
D = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3