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Properties

Label 39T11
Degree $39$
Order $156$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_{13}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 13: $F_{13}$

Low degree siblings

13T6, 26T8

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

Group invariants

Order:  $156=2^{2} \cdot 3 \cdot 13$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [156, 7]
Character table:    
      2  2  2  2  2   2   2   2   2   2   2   2   2   .
      3  1  1  1  1   1   1   1   1   1   1   1   1   .
     13  1  .  .  .   .   .   .   .   .   .   .   .   1

        1a 4a 4b 2a 12a  3a  6a 12b  6b 12c 12d  3b 13a
     2P 1a 2a 2a 1a  6b  3b  3b  6b  3a  6a  6a  3a 13a
     3P 1a 4b 4a 2a  4b  1a  2a  4a  2a  4b  4a  1a 13a
     5P 1a 4a 4b 2a 12c  3b  6b 12d  6a 12a 12b  3a 13a
     7P 1a 4b 4a 2a 12b  3a  6a 12a  6b 12d 12c  3b 13a
    11P 1a 4b 4a 2a 12d  3b  6b 12c  6a 12b 12a  3a 13a
    13P 1a 4a 4b 2a 12a  3a  6a 12b  6b 12c 12d  3b  1a

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(3)
  = (1-Sqrt(-3))/2 = -b3
C = -E(12)^7