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Properties

Label 39T43
Degree $39$
Order $5616$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\PSL(3,3)$

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Show commands: Magma

magma: G := TransitiveGroup(39, 43);
 

Group action invariants

Degree $n$:  $39$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(3,3)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8,18,13,5,20,37,36)(2,9,16,14,4,19,39,34)(3,7,17,15,6,21,38,35)(10,24,33,27)(11,23,32,26,12,22,31,25)(28,30), (1,37,25,7)(2,39,26,9,3,38,27,8)(4,13,22,35,30,19,33,18)(5,14,23,36,28,20,31,16)(6,15,24,34,29,21,32,17)(11,12)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 13: $\PSL(3,3)$

Low degree siblings

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

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 13, 13, 13 $ $432$ $13$ $( 1,20,13,29, 4,17,24, 8,32,38,35,10,27)( 2,21,14,30, 6,18,23, 9,33,39,34,12, 26)( 3,19,15,28, 5,16,22, 7,31,37,36,11,25)$
$ 13, 13, 13 $ $432$ $13$ $( 1,32,29,10,24,20,38, 4,27, 8,13,35,17)( 2,33,30,12,23,21,39, 6,26, 9,14,34, 18)( 3,31,28,11,22,19,37, 5,25, 7,15,36,16)$
$ 13, 13, 13 $ $432$ $13$ $( 1,27,10,35,38,32, 8,24,17, 4,29,13,20)( 2,26,12,34,39,33, 9,23,18, 6,30,14, 21)( 3,25,11,36,37,31, 7,22,16, 5,28,15,19)$
$ 13, 13, 13 $ $432$ $13$ $( 1,17,35,13, 8,27, 4,38,20,24,10,29,32)( 2,18,34,14, 9,26, 6,39,21,23,12,30, 33)( 3,16,36,15, 7,25, 5,37,19,22,11,28,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $117$ $2$ $( 1,25)( 2,27)( 3,26)( 4,24)( 5,23)( 6,22)( 8, 9)(11,12)(13,21)(14,20)(15,19) (17,18)(28,31)(29,33)(30,32)(34,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $104$ $3$ $( 1,15,23)( 2,13,24)( 3,14,22)( 4,27,21)( 5,25,19)( 6,26,20)( 7,10,36) ( 8,12,35)( 9,11,34)(28,29,30)(31,33,32)(37,38,39)$
$ 6, 6, 6, 6, 6, 3, 3, 2, 1 $ $936$ $6$ $( 1, 5,15,25,23,19)( 2, 4,13,27,24,21)( 3, 6,14,26,22,20)( 7,36,10) ( 8,34,12, 9,35,11)(17,18)(28,32,29,31,30,33)(37,39,38)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ $702$ $4$ $( 1,39,18,36)( 2,37,17,35)( 3,38,16,34)( 4,14, 6,13)( 5,15)( 7,32) ( 8,31, 9,33)(10,20,26,29)(11,21,25,28)(12,19,27,30)$
$ 8, 8, 8, 8, 4, 2, 1 $ $702$ $8$ $( 1,20,39,26,18,29,36,10)( 2,19,37,27,17,30,35,12)( 3,21,38,25,16,28,34,11) ( 4,33,14, 8, 6,31,13, 9)( 5,32,15, 7)(22,23)$
$ 8, 8, 8, 8, 4, 2, 1 $ $702$ $8$ $( 1,10,36,29,18,26,39,20)( 2,12,35,30,17,27,37,19)( 3,11,34,28,16,25,38,21) ( 4, 9,13,31, 6, 8,14,33)( 5, 7,15,32)(22,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $624$ $3$ $( 1,30,31)( 2,29,32)( 3,28,33)( 4,18,21)( 5,17,20)( 6,16,19)( 7,15,22) ( 8,14,24)( 9,13,23)(10,38,36)(11,37,34)(12,39,35)(25,26,27)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $5616=2^{4} \cdot 3^{3} \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  5616.a
magma: IdentifyGroup(G);
 
Character table:    
      2  4  4  1  1   .   .   .   .  .  3  3  3
      3  3  1  3  1   .   .   .   .  2  .  .  .
     13  1  .  .  .   1   1   1   1  .  .  .  .

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

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

magma: CharacterTable(G);