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

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);
 
Label:  5040.w
magma: IdentifyGroup(G);
 
Character table:    
      2  4  1  4   1  4  3  4  3  3  2  3  1  1   2  .
      3  2  .  1   .  1  .  1  2  1  1  1  2  1   1  .
      5  1  1  1   1  .  .  .  .  .  .  .  .  .   .  .
      7  1  .  .   .  .  .  .  .  .  .  .  .  .   .  1

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

magma: CharacterTable(G);