Properties

Label 9T34
Degree $9$
Order $362880$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_9$

Related objects

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Group action invariants

Degree $n$:  $9$
Transitive number $t$:  $34$
Group:  $S_9$
CHM label:  $S9$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,2), (1,2,3,4,5,6,7,8,9)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Low degree siblings

18T887, 36T28590

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1 $ $945$ $2$ $(2,4)(3,8)(5,6)(7,9)$
$ 4, 4, 1 $ $11340$ $4$ $(2,7,4,9)(3,5,8,6)$
$ 8, 1 $ $45360$ $8$ $(2,6,7,3,4,5,9,8)$
$ 2, 2, 1, 1, 1, 1, 1 $ $378$ $2$ $(3,4)(5,9)$
$ 3, 1, 1, 1, 1, 1, 1 $ $168$ $3$ $(1,8,2)$
$ 4, 2, 1, 1, 1 $ $7560$ $4$ $(3,5,4,9)(6,7)$
$ 3, 2, 2, 1, 1 $ $7560$ $6$ $(1,2,8)(3,4)(5,9)$
$ 4, 3, 2 $ $15120$ $12$ $(1,8,2)(3,9,4,5)(6,7)$
$ 2, 1, 1, 1, 1, 1, 1, 1 $ $36$ $2$ $(4,8)$
$ 7, 1, 1 $ $25920$ $7$ $(1,3,9,2,7,5,6)$
$ 7, 2 $ $25920$ $14$ $(1,7,3,5,9,6,2)(4,8)$
$ 4, 1, 1, 1, 1, 1 $ $756$ $4$ $(1,9,5,3)$
$ 4, 3, 1, 1 $ $15120$ $12$ $(1,3,5,9)(4,7,8)$
$ 4, 2, 2, 1 $ $11340$ $4$ $(1,2)(3,5,4,9)(6,7)$
$ 3, 3, 1, 1, 1 $ $3360$ $3$ $(2,5,7)(4,8,6)$
$ 3, 3, 2, 1 $ $10080$ $6$ $(2,7,5)(3,9)(4,6,8)$
$ 2, 2, 2, 1, 1, 1 $ $1260$ $2$ $(3,4)(5,7)(6,9)$
$ 3, 3, 3 $ $2240$ $3$ $(1,8,2)(3,5,9)(4,7,6)$
$ 6, 3 $ $20160$ $6$ $(1,2,8)(3,6,5,4,9,7)$
$ 3, 2, 1, 1, 1, 1 $ $2520$ $6$ $(2,5,7)(3,9)$
$ 3, 2, 2, 2 $ $2520$ $6$ $(1,5)(2,6)(3,9)(4,7,8)$
$ 6, 1, 1, 1 $ $10080$ $6$ $( 3, 6, 7, 8, 5, 9)$
$ 5, 1, 1, 1, 1 $ $3024$ $5$ $(1,9,6,7,5)$
$ 5, 2, 2 $ $9072$ $10$ $(1,7,9,5,6)(2,4)(3,8)$
$ 5, 4 $ $18144$ $20$ $(1,5,7,6,9)(2,8,4,3)$
$ 6, 2, 1 $ $30240$ $6$ $(2,4,5,8,7,6)(3,9)$
$ 5, 2, 1, 1 $ $18144$ $10$ $(1,9,8,7,6)(2,3)$
$ 9 $ $40320$ $9$ $( 1, 9, 6, 2, 5, 7, 8, 3, 4)$
$ 5, 3, 1 $ $24192$ $15$ $(1,8,2)(3,5,6,7,4)$

Group invariants

Order:  $362880=2^{7} \cdot 3^{4} \cdot 5 \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table: not available.