Properties

Label 8T50
Order \(40320\)
n \(8\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $S_8$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $50$
Group :  $S_8$
CHM label :  $S8$
Parity:  $-1$
Primitive:  Yes
Generators:  (1,2,3,4,5,6,7,8), (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

Degree 2: None

Degree 4: None

Low degree siblings

16T1838, 28T502, 30T1153, 35T44

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 $ $1$ $1$ $()$
$ 7, 1 $ $5760$ $7$ $(1,4,2,5,8,3,6)$
$ 2, 2, 1, 1, 1, 1 $ $210$ $2$ $(1,7)(2,8)$
$ 3, 1, 1, 1, 1, 1 $ $112$ $3$ $(3,6,5)$
$ 4, 1, 1, 1, 1 $ $420$ $4$ $(1,8,7,2)$
$ 3, 2, 2, 1 $ $1680$ $6$ $(1,7)(2,8)(3,5,6)$
$ 4, 3, 1 $ $3360$ $12$ $(1,2,7,8)(3,6,5)$
$ 2, 1, 1, 1, 1, 1, 1 $ $28$ $2$ $(2,8)$
$ 5, 1, 1, 1 $ $1344$ $5$ $(3,7,6,4,5)$
$ 5, 2, 1 $ $4032$ $10$ $(2,8)(3,4,7,5,6)$
$ 5, 3 $ $2688$ $15$ $(1,2,8)(3,5,4,6,7)$
$ 2, 2, 2, 2 $ $105$ $2$ $(1,4)(2,5)(3,8)(6,7)$
$ 3, 3, 1, 1 $ $1120$ $3$ $(1,7,3)(4,6,8)$
$ 6, 2 $ $3360$ $6$ $(1,8,7,4,3,6)(2,5)$
$ 2, 2, 2, 1, 1 $ $420$ $2$ $(1,4)(3,8)(6,7)$
$ 6, 1, 1 $ $3360$ $6$ $(1,6,3,4,7,8)$
$ 3, 3, 2 $ $1120$ $6$ $(1,3,7)(2,5)(4,8,6)$
$ 4, 4 $ $1260$ $4$ $( 1, 7, 4, 6)( 2, 3, 5, 8)$
$ 8 $ $5040$ $8$ $( 1, 3, 7, 5, 4, 8, 6, 2)$
$ 3, 2, 1, 1, 1 $ $1120$ $6$ $(1,5,6)(3,7)$
$ 4, 2, 1, 1 $ $2520$ $4$ $(1,2,7,8)(3,5)$
$ 4, 2, 2 $ $1260$ $4$ $(1,7)(2,8)(3,5,4,6)$

Group invariants

Order:  $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.