Properties

Label 39T37
Order \(2106\)
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 $t$ :  $37$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,37,30,2,39,28)(3,38,29)(4,8,16)(5,7,17,6,9,18)(10,26,32,11,25,33)(12,27,31)(13,36,19,14,35,20)(15,34,21)(22,24), (1,33,6,3,32,5,2,31,4)(7,12,20,8,10,21,9,11,19)(13,29,34)(14,30,35)(15,28,36)(16,38,23,18,37,22,17,39,24)(25,26,27)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
39:  $C_{13}:C_3$
78:  26T5

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 13: $C_{13}:C_3$

Low degree siblings

27T422

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 $ $26$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,12,11)(16,17,18)(19,21,20)(22,23,24) (31,32,33)(37,38,39)$
$ 13, 13, 13 $ $81$ $13$ $( 1,36,29,22,16,10, 6,37,32,25,21,13, 9)( 2,34,30,23,17,11, 4,38,33,26,19,14, 7)( 3,35,28,24,18,12, 5,39,31,27,20,15, 8)$
$ 13, 13, 13 $ $81$ $13$ $( 1,29,16, 6,32,21, 9,36,22,10,37,25,13)( 2,30,17, 4,33,19, 7,34,23,11,38,26, 14)( 3,28,18, 5,31,20, 8,35,24,12,39,27,15)$
$ 13, 13, 13 $ $81$ $13$ $( 1,16,32, 9,22,37,13,29, 6,21,36,10,25)( 2,17,33, 7,23,38,14,30, 4,19,34,11, 26)( 3,18,31, 8,24,39,15,28, 5,20,35,12,27)$
$ 13, 13, 13 $ $81$ $13$ $( 1,32,22,13, 6,36,25,16, 9,37,29,21,10)( 2,33,23,14, 4,34,26,17, 7,38,30,19, 11)( 3,31,24,15, 5,35,27,18, 8,39,28,20,12)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $117$ $3$ $( 4,29,12)( 5,30,10)( 6,28,11)( 7,18,20)( 8,16,21)( 9,17,19)(13,32,39) (14,33,37)(15,31,38)(22,36,26)(23,34,27)(24,35,25)$
$ 9, 9, 9, 3, 3, 3, 3 $ $234$ $9$ $( 1, 2, 3)( 4,29,11)( 5,30,12)( 6,28,10)( 7,16,20, 9,18,19, 8,17,21) (13,33,38,15,32,37,14,31,39)(22,36,26,23,34,27,24,35,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $117$ $3$ $( 4,12,29)( 5,10,30)( 6,11,28)( 7,20,18)( 8,21,16)( 9,19,17)(13,39,32) (14,37,33)(15,38,31)(22,26,36)(23,27,34)(24,25,35)$
$ 9, 9, 9, 3, 3, 3, 3 $ $234$ $9$ $( 1, 2, 3)( 4,11,28)( 5,12,29)( 6,10,30)( 7,19,18, 9,21,17, 8,20,16) (13,37,31,15,39,33,14,38,32)(22,26,36,23,27,34,24,25,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(13,15)(17,18)(19,20)(23,24)(25,26)(29,30)(31,32) (34,35)(37,38)$
$ 26, 13 $ $81$ $26$ $( 1,36,29,23,18,11, 5,37,33,26,21,13, 8)( 2,35,30,22,16,10, 6,39,31,25,19,15, 9, 3,34,28,24,17,12, 4,38,32,27,20,14, 7)$
$ 26, 13 $ $81$ $26$ $( 1,29,17, 5,32,20, 7,34,24,11,39,27,15)( 2,28,18, 4,33,19, 8,36,22,10,37,26, 13, 3,30,16, 6,31,21, 9,35,23,12,38,25,14)$
$ 26, 13 $ $81$ $26$ $( 1,16,32, 8,22,37,14,30, 6,20,34,12,26)( 2,18,33, 7,23,39,15,29, 4,19,35,11, 27, 3,17,31, 9,24,38,13,28, 5,21,36,10,25)$
$ 26, 13 $ $81$ $26$ $( 1,32,24,14, 4,34,27,18, 7,38,29,19,12, 2,31,22,13, 5,36,25,17, 8,37,30,21,10 )( 3,33,23,15, 6,35,26,16, 9,39,28,20,11)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ $351$ $6$ $( 2, 3)( 4,29,10, 5,28,11)( 6,30,12)( 7,18,19)( 8,17,20, 9,16,21)(13,31,39) (14,33,37,15,32,38)(22,36,26,24,34,25)(23,35,27)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ $351$ $6$ $( 2, 3)( 4,12,28, 6,10,30)( 5,11,29)( 7,20,17)( 8,19,18, 9,21,16) (13,38,33,14,37,31)(15,39,32)(22,26,35,23,25,36)(24,27,34)$

Group invariants

Order:  $2106=2 \cdot 3^{4} \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table:   
      2  1  .   1   1   1   1  1   .  1   .  1   1   1   1   1   1   1
      3  4  4   .   .   .   .  2   2  2   2  1   .   .   .   .   1   1
     13  1  .   1   1   1   1  .   .  .   .  1   1   1   1   1   .   .

        1a 3a 13a 13b 13c 13d 3b  9a 3c  9b 2a 26a 26b 26c 26d  6a  6b
     2P 1a 3a 13b 13c 13d 13a 3c  9b 3b  9a 1a 13b 13c 13d 13a  3c  3b
     3P 1a 1a 13a 13b 13c 13d 1a  3a 1a  3a 2a 26a 26b 26c 26d  2a  2a
     5P 1a 3a 13b 13c 13d 13a 3c  9b 3b  9a 2a 26b 26c 26d 26a  6b  6a
     7P 1a 3a 13d 13a 13b 13c 3b  9a 3c  9b 2a 26d 26a 26b 26c  6a  6b
    11P 1a 3a 13d 13a 13b 13c 3c  9b 3b  9a 2a 26d 26a 26b 26c  6b  6a
    13P 1a 3a  1a  1a  1a  1a 3b  9a 3c  9b 2a  2a  2a  2a  2a  6a  6b
    17P 1a 3a 13c 13d 13a 13b 3c  9b 3b  9a 2a 26c 26d 26a 26b  6b  6a
    19P 1a 3a 13b 13c 13d 13a 3b  9a 3c  9b 2a 26b 26c 26d 26a  6a  6b
    23P 1a 3a 13c 13d 13a 13b 3c  9b 3b  9a 2a 26c 26d 26a 26b  6b  6a

X.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
X.3      1  1   1   1   1   1  C   C /C  /C -1  -1  -1  -1  -1  -C -/C
X.4      1  1   1   1   1   1 /C  /C  C   C -1  -1  -1  -1  -1 -/C  -C
X.5      1  1   1   1   1   1  C   C /C  /C  1   1   1   1   1   C  /C
X.6      1  1   1   1   1   1 /C  /C  C   C  1   1   1   1   1  /C   C
X.7      3  3   A   B  /A  /B  .   .  .   .  3   A   B  /A  /B   .   .
X.8      3  3  /A  /B   A   B  .   .  .   .  3  /A  /B   A   B   .   .
X.9      3  3   B  /A  /B   A  .   .  .   .  3   B  /A  /B   A   .   .
X.10     3  3  /B   A   B  /A  .   .  .   .  3  /B   A   B  /A   .   .
X.11     3  3   A   B  /A  /B  .   .  .   . -3  -A  -B -/A -/B   .   .
X.12     3  3  /A  /B   A   B  .   .  .   . -3 -/A -/B  -A  -B   .   .
X.13     3  3   B  /A  /B   A  .   .  .   . -3  -B -/A -/B  -A   .   .
X.14     3  3  /B   A   B  /A  .   .  .   . -3 -/B  -A  -B -/A   .   .
X.15    26 -1   .   .   .   .  2  -1  2  -1  .   .   .   .   .   .   .
X.16    26 -1   .   .   .   .  D  -C /D -/C  .   .   .   .   .   .   .
X.17    26 -1   .   .   .   . /D -/C  D  -C  .   .   .   .   .   .   .

A = E(13)+E(13)^3+E(13)^9
B = E(13)^2+E(13)^5+E(13)^6
C = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
D = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3