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, 36T28590Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy Classes
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 1, 1, 1, 1, 1 $ | $378$ | $2$ | $(3,5)(4,6)$ |
| $ 5, 1, 1, 1, 1 $ | $3024$ | $5$ | $(1,7,9,2,8)$ |
| $ 5, 2, 2 $ | $9072$ | $10$ | $(1,2,7,8,9)(3,5)(4,6)$ |
| $ 4, 1, 1, 1, 1, 1 $ | $756$ | $4$ | $(3,6,5,4)$ |
| $ 5, 4 $ | $18144$ | $20$ | $(1,7,9,2,8)(3,6,5,4)$ |
| $ 2, 2, 2, 2, 1 $ | $945$ | $2$ | $(1,4)(2,5)(6,7)(8,9)$ |
| $ 4, 4, 1 $ | $11340$ | $4$ | $(1,7,4,6)(2,8,5,9)$ |
| $ 8, 1 $ | $45360$ | $8$ | $(1,8,7,5,4,9,6,2)$ |
| $ 3, 1, 1, 1, 1, 1, 1 $ | $168$ | $3$ | $(1,5,8)$ |
| $ 3, 2, 2, 1, 1 $ | $7560$ | $6$ | $(1,8,5)(4,9)(6,7)$ |
| $ 2, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $2$ | $(2,4)$ |
| $ 5, 2, 1, 1 $ | $18144$ | $10$ | $(2,4)(5,8,6,9,7)$ |
| $ 3, 2, 1, 1, 1, 1 $ | $2520$ | $6$ | $(1,8,5)(6,7)$ |
| $ 4, 2, 1, 1, 1 $ | $7560$ | $4$ | $(2,3)(4,7,9,6)$ |
| $ 4, 3, 2 $ | $15120$ | $12$ | $(1,5,8)(2,3)(4,6,9,7)$ |
| $ 4, 3, 1, 1 $ | $15120$ | $12$ | $(1,5,8)(4,7,9,6)$ |
| $ 3, 3, 1, 1, 1 $ | $3360$ | $3$ | $(2,4,9)(3,7,6)$ |
| $ 5, 3, 1 $ | $24192$ | $15$ | $(1,8,5)(2,4,6,3,7)$ |
| $ 6, 2, 1 $ | $30240$ | $6$ | $(2,3,7,5,8,9)(4,6)$ |
| $ 2, 2, 2, 1, 1, 1 $ | $1260$ | $2$ | $(2,5)(3,8)(7,9)$ |
| $ 6, 1, 1, 1 $ | $10080$ | $6$ | $(2,9,8,5,7,3)$ |
| $ 3, 3, 2, 1 $ | $10080$ | $6$ | $(1,4)(2,7,8)(3,5,9)$ |
| $ 4, 2, 2, 1 $ | $11340$ | $4$ | $( 1, 4)( 2, 5)( 6, 8, 7, 9)$ |
| $ 7, 1, 1 $ | $25920$ | $7$ | $(1,5,4,8,2,7,9)$ |
| $ 3, 3, 3 $ | $2240$ | $3$ | $(1,7,2)(3,9,6)(4,5,8)$ |
| $ 9 $ | $40320$ | $9$ | $(1,6,5,7,3,8,2,9,4)$ |
| $ 3, 2, 2, 2 $ | $2520$ | $6$ | $(1,5,8)(2,3)(4,6)(7,9)$ |
| $ 6, 3 $ | $20160$ | $6$ | $( 1, 3, 2, 6, 7, 9)( 4, 5, 8)$ |
| $ 7, 2 $ | $25920$ | $14$ | $(1,5,4,8,2,7,9)(3,6)$ |
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. |