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

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  1  1  .  4  3  3  3   2  4  2  1   1  3
      3  2  1  2  1  .  1  2  1  1   1  1  1  .   .  .
      5  1  .  .  .  .  .  .  .  .   .  1  .  1   1  .
      7  1  .  .  .  1  .  .  .  .   .  .  .  .   .  .

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

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