Properties

Label 39T9
Order \(156\)
n \(39\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{13}.S_3$

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Group action invariants

Degree $n$ :  $39$
Transitive number $t$ :  $9$
Group :  $D_{13}.S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,26,37,12,22,36,9,19,31,6,16,29,2,15,27,38,10,23,34,7,20,32,4,17,30,3,13,25,39,11,24,35,8,21,33,5,18,28), (1,20,31,13)(2,19,32,15)(3,21,33,14)(4,34,28,38)(5,36,29,37)(6,35,30,39)(7,12,26,24)(8,11,27,23)(9,10,25,22)(16,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$
52:  $C_{13}:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 13: $C_{13}:C_4$

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

Group invariants

Order:  $156=2^{2} \cdot 3 \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [156, 10]
Character table:   
      2  2  2  2  2  1  1   .   .   .   .   .   .   .   .   .
      3  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

        1a 2a 4a 4b 3a 6a 39a 13a 39b 39c 13b 39d 13c 39e 39f
     2P 1a 1a 2a 2a 3a 3a 39d 13b 39c 39e 13c 39f 13a 39b 39a
     3P 1a 2a 4b 4a 1a 2a 13b 13b 13b 13c 13c 13c 13a 13a 13a
     5P 1a 2a 4a 4b 3a 6a 39a 13a 39b 39c 13b 39d 13c 39e 39f
     7P 1a 2a 4b 4a 3a 6a 39e 13c 39f 39a 13a 39b 13b 39d 39c
    11P 1a 2a 4b 4a 3a 6a 39d 13b 39c 39e 13c 39f 13a 39b 39a
    13P 1a 2a 4a 4b 3a 6a  3a  1a  3a  3a  1a  3a  1a  3a  3a
    17P 1a 2a 4a 4b 3a 6a 39e 13c 39f 39a 13a 39b 13b 39d 39c
    19P 1a 2a 4b 4a 3a 6a 39e 13c 39f 39a 13a 39b 13b 39d 39c
    23P 1a 2a 4b 4a 3a 6a 39c 13b 39d 39f 13c 39e 13a 39a 39b
    29P 1a 2a 4a 4b 3a 6a 39c 13b 39d 39f 13c 39e 13a 39a 39b
    31P 1a 2a 4b 4a 3a 6a 39b 13a 39a 39d 13b 39c 13c 39f 39e
    37P 1a 2a 4a 4b 3a 6a 39c 13b 39d 39f 13c 39e 13a 39a 39b

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      1 -1  A -A  1 -1   1   1   1   1   1   1   1   1   1
X.4      1 -1 -A  A  1 -1   1   1   1   1   1   1   1   1   1
X.5      2 -2  .  . -1  1  -1   2  -1  -1   2  -1   2  -1  -1
X.6      2  2  .  . -1 -1  -1   2  -1  -1   2  -1   2  -1  -1
X.7      4  .  .  .  4  .   B   B   B   D   D   D   C   C   C
X.8      4  .  .  .  4  .   C   C   C   B   B   B   D   D   D
X.9      4  .  .  .  4  .   D   D   D   C   C   C   B   B   B
X.10     4  .  .  . -2  .   E   B  /E  /G   D   G   C  /F   F
X.11     4  .  .  . -2  .   F   C  /F  /E   B   E   D  /G   G
X.12     4  .  .  . -2  .   G   D  /G  /F   C   F   B  /E   E
X.13     4  .  .  . -2  .  /G   D   G   F   C  /F   B   E  /E
X.14     4  .  .  . -2  .  /E   B   E   G   D  /G   C   F  /F
X.15     4  .  .  . -2  .  /F   C   F   E   B  /E   D   G  /G

A = -E(4)
  = -Sqrt(-1) = -i
B = E(13)+E(13)^5+E(13)^8+E(13)^12
C = E(13)^4+E(13)^6+E(13)^7+E(13)^9
D = E(13)^2+E(13)^3+E(13)^10+E(13)^11
E = E(39)^23+E(39)^28+E(39)^29+E(39)^37
F = E(39)^14+E(39)^31+E(39)^34+E(39)^38
G = E(39)^7+E(39)^17+E(39)^19+E(39)^35