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
Nilpotency class:  $-1$ (not nilpotent)
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 $ $1$ $1$ $()$
$ 7 $ $720$ $7$ $(1,6,7,3,2,4,5)$
$ 2, 2, 1, 1, 1 $ $105$ $2$ $(2,7)(3,4)$
$ 3, 1, 1, 1, 1 $ $70$ $3$ $(1,5,6)$
$ 4, 1, 1, 1 $ $210$ $4$ $(2,3,7,4)$
$ 3, 2, 2 $ $210$ $6$ $(1,6,5)(2,7)(3,4)$
$ 4, 3 $ $420$ $12$ $(1,5,6)(2,4,7,3)$
$ 2, 2, 2, 1 $ $105$ $2$ $(1,6)(2,3)(5,7)$
$ 3, 3, 1 $ $280$ $3$ $(1,3,5)(2,7,6)$
$ 6, 1 $ $840$ $6$ $(1,7,3,6,5,2)$
$ 2, 1, 1, 1, 1, 1 $ $21$ $2$ $(2,7)$
$ 3, 2, 1, 1 $ $420$ $6$ $(1,5,6)(2,7)$
$ 4, 2, 1 $ $630$ $4$ $( 1, 3, 6, 2)( 5, 7)$
$ 5, 1, 1 $ $504$ $5$ $(1,3,4,5,6)$
$ 5, 2 $ $504$ $10$ $(1,4,6,3,5)(2,7)$

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

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

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