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Properties

Label 13T7
Degree $13$
Order $5616$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(3,3)$

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Group action invariants

Degree $n$:  $13$
Transitive number $t$:  $7$
Group:  $\PSL(3,3)$
CHM label:  $L(13)=PSL(3,3)$
Parity:  $1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$\card{\Aut(F/K)}$:  $1$
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,12)(4,11)(5,6)(7,10)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

13T7, 26T39 x 2, 39T43 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $117$ $2$ $( 3, 8)( 5,13)( 6, 9)(11,12)$
$ 3, 3, 3, 1, 1, 1, 1 $ $104$ $3$ $( 1, 2, 4)( 5,12, 9)( 6,13,11)$
$ 6, 3, 2, 1, 1 $ $936$ $6$ $( 1, 4, 2)( 3, 8)( 5, 6,12,13, 9,11)$
$ 13 $ $432$ $13$ $( 1,10, 5, 7,11, 6, 8, 3, 2, 9, 4,12,13)$
$ 13 $ $432$ $13$ $( 1, 2, 7,12, 8,10, 9,11,13, 3, 5, 4, 6)$
$ 13 $ $432$ $13$ $( 1,13,12, 4, 9, 2, 3, 8, 6,11, 7, 5,10)$
$ 13 $ $432$ $13$ $( 1, 6, 4, 5, 3,13,11, 9,10, 8,12, 7, 2)$
$ 4, 4, 2, 2, 1 $ $702$ $4$ $( 1, 5)( 2,12, 3, 7)( 4,10, 9,13)( 6, 8)$
$ 8, 4, 1 $ $702$ $8$ $( 1, 6, 5, 8)( 2, 4,12,10, 3, 9, 7,13)$
$ 8, 4, 1 $ $702$ $8$ $( 1, 8, 5, 6)( 2,13, 7, 9, 3,10,12, 4)$
$ 3, 3, 3, 3, 1 $ $624$ $3$ $( 1,11, 5)( 2, 6,12)( 3, 8, 7)( 4,13, 9)$

Group invariants

Order:  $5616=2^{4} \cdot 3^{3} \cdot 13$
Cyclic:  no
Abelian:  no
Solvable:  no
Label:  not available
Character table:    
      2  4   .   .   .   .  4  3  3  3  1  1  .
      3  3   .   .   .   .  1  .  .  .  3  1  2
     13  1   1   1   1   1  .  .  .  .  .  .  .

        1a 13a 13b 13c 13d 2a 4a 8a 8b 3a 6a 3b
     2P 1a 13d 13a 13b 13c 1a 2a 4a 4a 3a 3a 3b
     3P 1a 13a 13b 13c 13d 2a 4a 8a 8b 1a 2a 1a
     5P 1a 13d 13a 13b 13c 2a 4a 8b 8a 3a 6a 3b
     7P 1a 13b 13c 13d 13a 2a 4a 8b 8a 3a 6a 3b
    11P 1a 13b 13c 13d 13a 2a 4a 8a 8b 3a 6a 3b
    13P 1a  1a  1a  1a  1a 2a 4a 8b 8a 3a 6a 3b

X.1      1   1   1   1   1  1  1  1  1  1  1  1
X.2     12  -1  -1  -1  -1  4  .  .  .  3  1  .
X.3     13   .   .   .   . -3  1 -1 -1  4  .  1
X.4     16   A  /B  /A   B  .  .  .  . -2  .  1
X.5     16  /A   B   A  /B  .  .  .  . -2  .  1
X.6     16   B   A  /B  /A  .  .  .  . -2  .  1
X.7     16  /B  /A   B   A  .  .  .  . -2  .  1
X.8     26   .   .   .   .  2  2  .  . -1 -1 -1
X.9     26   .   .   .   . -2  .  C -C -1  1 -1
X.10    26   .   .   .   . -2  . -C  C -1  1 -1
X.11    27   1   1   1   1  3 -1 -1 -1  .  .  .
X.12    39   .   .   .   . -1 -1  1  1  3 -1  .

A = E(13)^2+E(13)^5+E(13)^6
B = E(13)^4+E(13)^10+E(13)^12
C = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2