# Properties

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

# Related objects

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

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);