Properties

Label 7T7
Order \(5040\)
n \(7\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $S_7$

Related objects

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Group action invariants

Degree $n$ :  $7$
Transitive number $t$ :  $7$
Group :  $S_7$
CHM label :  $S7$
Parity:  $-1$
Primitive:  Yes
Generators:  (1,2,3,4,5,6,7), (1,2)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

14T46, 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 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1, 1 $ $21$ $2$ $(2,3)$
$ 3, 1, 1, 1, 1 $ $70$ $3$ $(1,5,7)$
$ 3, 2, 1, 1 $ $420$ $6$ $(1,7,5)(2,3)$
$ 3, 3, 1 $ $280$ $3$ $(1,5,7)(2,4,6)$
$ 2, 2, 1, 1, 1 $ $105$ $2$ $(1,7)(2,5)$
$ 4, 2, 1 $ $630$ $4$ $(1,2,7,5)(3,4)$
$ 2, 2, 2, 1 $ $105$ $2$ $(1,7)(2,5)(3,4)$
$ 4, 1, 1, 1 $ $210$ $4$ $(1,5,7,2)$
$ 3, 2, 2 $ $210$ $6$ $(1,7)(2,5)(3,6,4)$
$ 4, 3 $ $420$ $12$ $(1,2,7,5)(3,4,6)$
$ 6, 1 $ $840$ $6$ $(1,6,5,2,7,4)$
$ 7 $ $720$ $7$ $(1,5,6,7,3,4,2)$
$ 5, 1, 1 $ $504$ $5$ $(1,7,6,3,5)$
$ 5, 2 $ $504$ $10$ $(1,3,7,5,6)(2,4)$

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

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

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