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Properties

Label 14T46
Order \(5040\)
n \(14\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $46$
CHM label :  $2[1/2]S(7)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $2$

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

Group invariants

Order:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2  4  4  3  3  3   2  .  4  4  2  1  1  3  1   1
      3  2  1  2  1  1   1  .  1  1  1  2  1  .  .   .
      5  1  .  .  .  .   .  .  .  1  .  .  .  .  1   1
      7  1  .  .  .  .   .  1  .  .  .  .  .  .  .   .

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

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