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Properties

Label 7T7
Degree $7$
Order $5040$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_7$

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Show commands: Magma

magma: G := TransitiveGroup(7, 7);
 

Group action invariants

Degree $n$:  $7$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_7$
CHM label:  $S7$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6,7), (1,2)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  5040.w
magma: IdentifyGroup(G);
 
Character table:    
      2  4  4  3  2  4  3  3   2  4  3  1  1  1   1  .
      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 2c 4b 3b 6c 5a 10a 7a
     2P 1a 1a 3a 3a 1a 2b 3a  6b 1a 2b 3b 3b 5a  5a 7a
     3P 1a 2a 1a 2a 2b 4a 2b  4a 2c 4b 1a 2c 5a 10a 7a
     5P 1a 2a 3a 6a 2b 4a 6b 12a 2c 4b 3b 6c 1a  2a 7a
     7P 1a 2a 3a 6a 2b 4a 6b 12a 2c 4b 3b 6c 5a 10a 1a

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   . -2  . -1  1 -1  -1  .
X.4     14 -4 -1 -1  2  2 -1  -1  .  .  2  . -1   1  .
X.5     15 -5  3  1 -1 -1 -1  -1  3 -1  .  .  .   .  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  3 -1  .  .  1  -1  .
X.8     21  1 -3  1  1 -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  4 -1  1  2 -2 -1   1  .  .  2  . -1  -1  .
X.12    15  5  3 -1 -1  1 -1   1 -3 -1  .  .  .   .  1
X.13    14  6  2  .  2  .  2   .  2  . -1 -1 -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

magma: CharacterTable(G);