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
Nilpotency class:  $-1$ (not nilpotent)
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 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1, 1, 1 $ $28$ $2$ $(1,5)$
$ 3, 1, 1, 1, 1, 1 $ $112$ $3$ $(2,6,3)$
$ 3, 2, 1, 1, 1 $ $1120$ $6$ $(1,5)(2,3,6)$
$ 5, 1, 1, 1 $ $1344$ $5$ $(1,5,7,8,4)$
$ 5, 3 $ $2688$ $15$ $(1,7,4,5,8)(2,3,6)$
$ 5, 2, 1 $ $4032$ $10$ $(1,8,2,5,4)(3,7)$
$ 2, 2, 2, 1, 1 $ $420$ $2$ $(1,6)(4,7)(5,8)$
$ 3, 3, 1, 1 $ $1120$ $3$ $(1,4,5)(6,7,8)$
$ 6, 1, 1 $ $3360$ $6$ $(1,8,4,6,5,7)$
$ 2, 2, 2, 2 $ $105$ $2$ $(1,6)(2,3)(4,7)(5,8)$
$ 6, 2 $ $3360$ $6$ $(1,7,5,6,4,8)(2,3)$
$ 3, 3, 2 $ $1120$ $6$ $(1,5,4)(2,3)(6,8,7)$
$ 2, 2, 1, 1, 1, 1 $ $210$ $2$ $( 1, 6)( 5, 8)$
$ 4, 2, 1, 1 $ $2520$ $4$ $( 1, 8, 6, 5)( 4, 7)$
$ 4, 1, 1, 1, 1 $ $420$ $4$ $( 1, 5, 6, 8)$
$ 4, 4 $ $1260$ $4$ $( 1, 5, 6, 8)( 2, 7, 4, 3)$
$ 4, 2, 2 $ $1260$ $4$ $( 1, 5, 6, 8)( 2, 3)( 4, 7)$
$ 8 $ $5040$ $8$ $(1,5,7,3,8,6,2,4)$
$ 3, 2, 2, 1 $ $1680$ $6$ $(1,6)(2,4,3)(5,8)$
$ 4, 3, 1 $ $3360$ $12$ $(1,5,6,8)(2,3,4)$
$ 7, 1 $ $5760$ $7$ $(1,3,6,8,7,2,4)$

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.