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Properties

Label 28T120
Order \(1092\)
n \(28\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $\PSL(2,13)$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $120$
Group :  $\PSL(2,13)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,26,11,15,17,13,10)(2,25,12,16,18,14,9)(3,7,5,28,20,22,23)(4,8,6,27,19,21,24), (1,14,9,17,28,3,20)(2,13,10,18,27,4,19)(5,7,26,15,23,22,11)(6,8,25,16,24,21,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: $\PSL(2,13)$

Low degree siblings

14T30, 42T176

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $91$ $2$ $( 1,15)( 2,16)( 3, 4)( 5,14)( 6,13)( 7,20)( 8,19)( 9,26)(10,25)(11,23)(12,24) (17,18)(21,28)(22,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $182$ $3$ $( 1,27,14)( 2,28,13)( 5,15,22)( 6,16,21)( 7,12, 9)( 8,11,10)(19,23,25) (20,24,26)$
$ 6, 6, 6, 6, 2, 2 $ $182$ $6$ $( 1, 5,27,15,14,22)( 2, 6,28,16,13,21)( 3, 4)( 7,26,12,20, 9,24) ( 8,25,11,19,10,23)(17,18)$
$ 7, 7, 7, 7 $ $156$ $7$ $( 1, 5, 9,14,22,17, 4)( 2, 6,10,13,21,18, 3)( 7,12,23,27,26,15,19) ( 8,11,24,28,25,16,20)$
$ 7, 7, 7, 7 $ $156$ $7$ $( 1,22, 5,17, 9, 4,14)( 2,21, 6,18,10, 3,13)( 7,26,12,15,23,19,27) ( 8,25,11,16,24,20,28)$
$ 7, 7, 7, 7 $ $156$ $7$ $( 1, 9,22, 4, 5,14,17)( 2,10,21, 3, 6,13,18)( 7,23,26,19,12,27,15) ( 8,24,25,20,11,28,16)$
$ 13, 13, 1, 1 $ $84$ $13$ $( 1, 9,12,21,24,20,14, 5, 4, 8,25,18,15)( 2,10,11,22,23,19,13, 6, 3, 7,26,17, 16)$
$ 13, 13, 1, 1 $ $84$ $13$ $( 1, 4,21,18,14, 9, 8,24,15, 5,12,25,20)( 2, 3,22,17,13,10, 7,23,16, 6,11,26, 19)$

Group invariants

Order:  $1092=2^{2} \cdot 3 \cdot 7 \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1092, 25]
Character table:    
     2  2   .   .  1  2  1  .  .  .
     3  1   .   .  1  1  1  .  .  .
     7  1   .   .  .  .  .  1  1  1
    13  1   1   1  .  .  .  .  .  .

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

X.1     1   1   1  1  1  1  1  1  1
X.2     7   A  *A  1 -1 -1  .  .  .
X.3     7  *A   A  1 -1 -1  .  .  .
X.4    12  -1  -1  .  .  .  B  C  D
X.5    12  -1  -1  .  .  .  C  D  B
X.6    12  -1  -1  .  .  .  D  B  C
X.7    13   .   .  1  1  1 -1 -1 -1
X.8    14   1   1 -1  2 -1  .  .  .
X.9    14   1   1 -1 -2  1  .  .  .

A = -E(13)-E(13)^3-E(13)^4-E(13)^9-E(13)^10-E(13)^12
  = (1-Sqrt(13))/2 = -b13
B = -E(7)^3-E(7)^4
C = -E(7)^2-E(7)^5
D = -E(7)-E(7)^6