Properties

Label 14T47
Order \(5040\)
n \(14\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_7\times C_2$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $47$
Group :  $A_7\times C_2$
CHM label :  $2[x]A(7)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,13,5)(6,12,10), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (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:  $C_2$
2520:  $A_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $A_7$

Low degree siblings

30T566 x 2, 42T409, 42T410

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 $ $1$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1, 9)( 2, 8)( 3,11)( 4,10)$
$ 2, 2, 2, 2, 2, 2, 2 $ $105$ $2$ $( 1, 2)( 3, 4)( 5,12)( 6,13)( 7,14)( 8, 9)(10,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 1, 9, 3)( 2,10, 8)$
$ 6, 2, 2, 2, 2 $ $70$ $6$ $( 1, 2, 3, 8, 9,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 3, 3, 2, 2, 2, 2 $ $210$ $6$ $( 1, 9, 3)( 2,10, 8)( 4,12)( 5,11)( 6,14)( 7,13)$
$ 6, 2, 2, 2, 2 $ $210$ $6$ $( 1, 2, 3, 8, 9,10)( 4, 5)( 6, 7)(11,12)(13,14)$
$ 3, 3, 3, 3, 1, 1 $ $280$ $3$ $( 1, 9, 3)( 2,10, 8)( 4,12, 6)( 5,13,11)$
$ 6, 6, 2 $ $280$ $6$ $( 1, 2, 3, 8, 9,10)( 4, 5, 6,11,12,13)( 7,14)$
$ 4, 4, 2, 2, 1, 1 $ $630$ $4$ $( 1, 9, 3,11)( 2,10, 4, 8)( 5,13)( 6,12)$
$ 4, 4, 2, 2, 2 $ $630$ $4$ $( 1, 2, 3, 4)( 5, 6)( 7,14)( 8, 9,10,11)(12,13)$
$ 5, 5, 1, 1, 1, 1 $ $504$ $5$ $( 1, 9, 3,11, 5)( 2,10, 4,12, 8)$
$ 10, 2, 2 $ $504$ $10$ $( 1, 2, 3, 4, 5, 8, 9,10,11,12)( 6,13)( 7,14)$
$ 7, 7 $ $360$ $7$ $( 1, 9, 3,11, 5,13, 7)( 2,10, 4,12, 6,14, 8)$
$ 14 $ $360$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 7, 7 $ $360$ $7$ $( 1, 9, 3,11, 5, 7,13)( 2,10, 4,12,14, 6, 8)$
$ 14 $ $360$ $14$ $( 1, 2, 3, 4, 5,14,13, 8, 9,10,11,12, 7, 6)$

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

        1a  2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 5a 10a 7a 14a 7b 14b
     2P 1a  1a 1a 1a 3a 3a 3a 3a 3b 3b 2b 2b 5a  5a 7a  7a 7b  7b
     3P 1a  2a 2b 2c 1a 2a 2b 2c 1a 2a 4a 4b 5a 10a 7b 14b 7a 14a
     5P 1a  2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 1a  2a 7b 14b 7a 14a
     7P 1a  2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 5a 10a 1a  2a 1a  2a
    11P 1a  2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 5a 10a 7a 14a 7b 14b
    13P 1a  2a 2b 2c 3a 6a 6b 6c 3b 6d 4a 4b 5a 10a 7b 14b 7a 14a

X.1      1   1  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  -1  1  -1
X.3      6   6  2  2  3  3 -1 -1  .  .  .  .  1   1 -1  -1 -1  -1
X.4      6  -6  2 -2  3 -3 -1  1  .  .  .  .  1  -1 -1   1 -1   1
X.5     10 -10 -2  2  1 -1  1 -1  1 -1  .  .  .   .  A  -A /A -/A
X.6     10 -10 -2  2  1 -1  1 -1  1 -1  .  .  .   . /A -/A  A  -A
X.7     10  10 -2 -2  1  1  1  1  1  1  .  .  .   .  A   A /A  /A
X.8     10  10 -2 -2  1  1  1  1  1  1  .  .  .   . /A  /A  A   A
X.9     14  14  2  2  2  2  2  2 -1 -1  .  . -1  -1  .   .  .   .
X.10    14  14  2  2 -1 -1 -1 -1  2  2  .  . -1  -1  .   .  .   .
X.11    14 -14  2 -2  2 -2  2 -2 -1  1  .  . -1   1  .   .  .   .
X.12    14 -14  2 -2 -1  1 -1  1  2 -2  .  . -1   1  .   .  .   .
X.13    15  15 -1 -1  3  3 -1 -1  .  . -1 -1  .   .  1   1  1   1
X.14    15 -15 -1  1  3 -3 -1  1  .  . -1  1  .   .  1  -1  1  -1
X.15    21  21  1  1 -3 -3  1  1  .  . -1 -1  1   1  .   .  .   .
X.16    21 -21  1 -1 -3  3  1 -1  .  . -1  1  1  -1  .   .  .   .
X.17    35  35 -1 -1 -1 -1 -1 -1 -1 -1  1  1  .   .  .   .  .   .
X.18    35 -35 -1  1 -1  1 -1  1 -1  1  1 -1  .   .  .   .  .   .

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7