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Properties

Label 14T46
Degree $14$
Order $5040$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_7$

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Show commands: Magma

magma: G := TransitiveGroup(14, 46);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $46$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_7$
CHM label:  $2[1/2]S(7)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8)(2,7)(3,10)(4,11)(5,12)(6,13)(9,14), (3,13,5)(6,12,10), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $S_7$

Low degree siblings

7T7, 21T38, 30T565, 35T31, 42T411, 42T412, 42T413, 42T418

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 2,12)( 3, 7)( 5, 9)(10,14)$
$ 4, 4, 2, 2, 2 $ $210$ $4$ $( 1, 8)( 2, 7,12, 3)( 4,11)( 5,10, 9,14)( 6,13)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 4)( 2, 9)( 3,10)( 5,12)( 6,13)( 7,14)( 8,11)$
$ 2, 2, 2, 2, 2, 2, 2 $ $105$ $2$ $( 1, 8)( 2, 7)( 3,12)( 4,13)( 5,10)( 6,11)( 9,14)$
$ 3, 3, 3, 3, 1, 1 $ $280$ $3$ $( 2, 6,10)( 3, 9,13)( 4,12,14)( 5, 7,11)$
$ 6, 6, 2 $ $840$ $6$ $( 1, 8)( 2,11, 6, 5,10, 7)( 3,14, 9, 4,13,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 4,12,14)( 5, 7,11)$
$ 4, 4, 2, 2, 1, 1 $ $630$ $4$ $( 1,13)( 2,10,12,14)( 3, 5, 7, 9)( 6, 8)$
$ 5, 5, 1, 1, 1, 1 $ $504$ $5$ $( 2,12,14, 6,10)( 3, 9, 5, 7,13)$
$ 10, 2, 2 $ $504$ $10$ $( 1, 4)( 2, 7,10, 5, 6, 9,14, 3,12,13)( 8,11)$
$ 6, 2, 2, 2, 2 $ $420$ $6$ $( 1, 2)( 3,10)( 4, 5,14,11,12, 7)( 6,13)( 8, 9)$
$ 3, 3, 2, 2, 2, 2 $ $210$ $6$ $( 1, 9)( 2, 8)( 3,13)( 4,14,12)( 5,11, 7)( 6,10)$
$ 6, 4, 4 $ $420$ $12$ $( 1, 6, 9,10)( 2, 3, 8,13)( 4, 7,12,11,14, 5)$
$ 7, 7 $ $720$ $7$ $( 1,11, 9, 7, 3,13, 5)( 2,14,10, 6,12, 8, 4)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  5040.w
magma: IdentifyGroup(G);
 
Character table:    
      2  4  4  1  1  4  3  3  3  3  4  2  1   1   2  .
      3  2  1  2  1  1  .  1  2  1  1  1  .   .   1  .
      5  1  .  .  .  .  .  .  .  .  1  .  1   1   .  .
      7  1  .  .  .  .  .  .  .  .  .  .  .   .   .  1

        1a 2a 3a 6a 2b 4a 4b 3b 6b 2c 6c 5a 10a 12a 7a
     2P 1a 1a 3a 3a 1a 2b 2b 3b 3b 1a 3b 5a  5a  6b 7a
     3P 1a 2a 1a 2a 2b 4a 4b 1a 2b 2c 2c 5a 10a  4b 7a
     5P 1a 2a 3a 6a 2b 4a 4b 3b 6b 2c 6c 1a  2c 12a 7a
     7P 1a 2a 3a 6a 2b 4a 4b 3b 6b 2c 6c 5a 10a 12a 1a
    11P 1a 2a 3a 6a 2b 4a 4b 3b 6b 2c 6c 5a 10a 12a 7a

X.1      1  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  -1  1
X.3      6  .  .  .  2  .  2  3 -1  4  1  1  -1  -1 -1
X.4      6  .  .  .  2  . -2  3 -1 -4 -1  1   1   1 -1
X.5     14  .  2  .  2  . -2 -1 -1  4  1 -1  -1   1  .
X.6     14 -2 -1  1  2  .  .  2  2 -6  . -1  -1   .  .
X.7     14  .  2  .  2  .  2 -1 -1 -4 -1 -1   1  -1  .
X.8     14  2 -1 -1  2  .  .  2  2  6  . -1   1   .  .
X.9     15  3  .  . -1 -1 -1  3 -1 -5  1  .   .  -1  1
X.10    15 -3  .  . -1 -1  1  3 -1  5 -1  .   .   1  1
X.11    20  .  2  . -4  .  .  2  2  .  .  .   .   . -1
X.12    21 -3  .  .  1 -1 -1 -3  1  1  1  1   1  -1  .
X.13    21  3  .  .  1 -1  1 -3  1 -1 -1  1  -1   1  .
X.14    35 -1 -1 -1 -1  1  1 -1 -1 -5  1  .   .   1  .
X.15    35  1 -1  1 -1  1 -1 -1 -1  5 -1  .   .  -1  .

magma: CharacterTable(G);