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Properties

Label 39T16
Order \(351\)
n \(39\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^3:C_{13}$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 13: $C_{13}$

Low degree siblings

27T134

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

Group invariants

Order:  $351=3^{3} \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [351, 12]
Character table:    
      3  3  3  3   .   .   .   .   .   .   .   .   .   .   .   .
     13  1  .  .   1   1   1   1   1   1   1   1   1   1   1   1

        1a 3a 3b 13a 13b 13c 13d 13e 13f 13g 13h 13i 13j 13k 13l
     2P 1a 3b 3a 13b 13d 13f 13h 13j 13l 13a 13c 13e 13g 13i 13k
     3P 1a 1a 1a 13c 13f 13i 13l 13b 13e 13h 13k 13a 13d 13g 13j
     5P 1a 3b 3a 13e 13j 13b 13g 13l 13d 13i 13a 13f 13k 13c 13h
     7P 1a 3a 3b 13g 13a 13h 13b 13i 13c 13j 13d 13k 13e 13l 13f
    11P 1a 3b 3a 13k 13i 13g 13e 13c 13a 13l 13j 13h 13f 13d 13b
    13P 1a 3a 3b  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a

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

A = -E(3)+2*E(3)^2
  = (-1-3*Sqrt(-3))/2 = -2-3b3
B = E(13)^12
C = E(13)^11
D = E(13)^10
E = E(13)^9
F = E(13)^8
G = E(13)^7