Properties

Label 39T14
Order \(234\)
n \(39\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{39}:C_3$

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Group action invariants

Degree $n$ :  $39$
Transitive number $t$ :  $14$
Group :  $D_{39}:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,34,12,33,38,24)(2,36,10,32,39,23)(3,35,11,31,37,22)(4,7,19,30,26,13)(5,9,20,29,27,15)(6,8,21,28,25,14)(17,18), (1,29,39)(2,30,37)(3,28,38)(4,16,7)(5,17,8)(6,18,9)(10,31,26)(11,32,27)(12,33,25)(13,19,36)(14,20,34)(15,21,35)(22,24,23)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
78:  $C_{13}:C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 13: $C_{13}:C_6$

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

Group invariants

Order:  $234=2 \cdot 3^{2} \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [234, 9]
Character table:   
      2  1  1  1   1   1  1  .   .   .   .   .   .   .   .   .
      3  2  2  2   1   1  1  2   2   2   1   1   1   1   1   1
     13  1  .  .   .   .  .  1   .   .   1   1   1   1   1   1

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

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  A /A -/A  -A -1  1   A  /A   1   1   1   1   1   1
X.4      1 /A  A  -A -/A -1  1  /A   A   1   1   1   1   1   1
X.5      1  A /A  /A   A  1  1   A  /A   1   1   1   1   1   1
X.6      1 /A  A   A  /A  1  1  /A   A   1   1   1   1   1   1
X.7      2  2  2   .   .  . -1  -1  -1  -1   2  -1   2  -1  -1
X.8      2  B /B   .   .  . -1 -/A  -A  -1   2  -1   2  -1  -1
X.9      2 /B  B   .   .  . -1  -A -/A  -1   2  -1   2  -1  -1
X.10     6  .  .   .   .  .  6   .   .   C   C   C  *C  *C  *C
X.11     6  .  .   .   .  .  6   .   .  *C  *C  *C   C   C   C
X.12     6  .  .   .   .  . -3   .   .   D   C   F  *C   G   E
X.13     6  .  .   .   .  . -3   .   .   E  *C   G   C   D   F
X.14     6  .  .   .   .  . -3   .   .   F   C   D  *C   E   G
X.15     6  .  .   .   .  . -3   .   .   G  *C   E   C   F   D

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)
  = -1+Sqrt(-3) = 2b3
C = E(13)^2+E(13)^5+E(13)^6+E(13)^7+E(13)^8+E(13)^11
  = (-1-Sqrt(13))/2 = -1-b13
D = E(39)^8+E(39)^11+E(39)^19+E(39)^20+E(39)^28+E(39)^31
E = E(39)^4+E(39)^10+E(39)^14+E(39)^25+E(39)^29+E(39)^35
F = E(39)^2+E(39)^5+E(39)^7+E(39)^32+E(39)^34+E(39)^37
G = E(39)+E(39)^16+E(39)^17+E(39)^22+E(39)^23+E(39)^38