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

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.