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

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

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

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  . -1 -2  1  .  2 -1  -1  .   .
X.4     14 -4 -1 -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 -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  .  2  . -4  .  2  .  .  .  2  .   . -1   .
X.10    35  5 -1 -1 -1  1 -1  1  1 -1 -1  .   .  .  -1
X.11    14  4 -1  1  2  .  2  .  . -2 -1 -1  -1  .   1
X.12    15  5  3 -1 -1 -1  . -3  .  1 -1  .   .  1   1
X.13    14  6  2  .  2  . -1  2 -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