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

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

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

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