Properties

Label 9T34
Order \(362880\)
n \(9\)
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)
Generators:  (1,2), (1,2,3,4,5,6,7,8,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

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

Group invariants

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