Group action invariants
| Degree $n$ : | $10$ | |
| Transitive number $t$ : | $45$ | |
| Group : | $S_{10}$ | |
| CHM label : | $S10$ | |
| Parity: | $-1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,2), (1,2,3,4,5,6,7,8,9,10) | |
| $|\Aut(F/K)|$: | $1$ |
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 5: None
Low degree siblings
20T1007, 45T2246Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 1, 1 $ | $4725$ | $2$ | $( 1,10)( 4, 7)( 5, 9)( 6, 8)$ |
| $ 4, 4, 1, 1 $ | $56700$ | $4$ | $( 1, 8,10, 6)( 4, 9, 7, 5)$ |
| $ 8, 1, 1 $ | $226800$ | $8$ | $( 1, 4, 8, 9,10, 7, 6, 5)$ |
| $ 8, 2 $ | $226800$ | $8$ | $( 1, 5, 6, 7,10, 9, 8, 4)( 2, 3)$ |
| $ 3, 3, 1, 1, 1, 1 $ | $8400$ | $3$ | $( 1, 4, 9)( 2,10, 7)$ |
| $ 6, 2, 1, 1 $ | $151200$ | $6$ | $( 1,10, 4, 7, 9, 2)( 6, 8)$ |
| $ 4, 4, 2 $ | $56700$ | $4$ | $( 1, 8, 4, 5)( 2, 3)( 6, 7, 9,10)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1 $ | $630$ | $2$ | $( 4, 7)( 6, 8)$ |
| $ 4, 1, 1, 1, 1, 1, 1 $ | $1260$ | $4$ | $( 4, 8, 7, 6)$ |
| $ 4, 2, 1, 1, 1, 1 $ | $18900$ | $4$ | $( 1, 5,10, 2)( 3, 9)$ |
| $ 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $45$ | $2$ | $( 3, 5)$ |
| $ 3, 1, 1, 1, 1, 1, 1, 1 $ | $240$ | $3$ | $( 1, 9, 4)$ |
| $ 3, 2, 1, 1, 1, 1, 1 $ | $5040$ | $6$ | $( 1, 4, 9)( 3, 5)$ |
| $ 2, 2, 2, 1, 1, 1, 1 $ | $3150$ | $2$ | $( 3, 5)( 6, 8)( 7,10)$ |
| $ 3, 2, 2, 2, 1 $ | $25200$ | $6$ | $( 1, 4, 9)( 3, 5)( 6, 8)( 7,10)$ |
| $ 3, 2, 2, 1, 1, 1 $ | $25200$ | $6$ | $( 1, 9, 4)( 3, 5)( 6, 8)$ |
| $ 3, 3, 3, 1 $ | $22400$ | $3$ | $( 1, 4, 9)( 3,10, 8)( 5, 7, 6)$ |
| $ 9, 1 $ | $403200$ | $9$ | $( 1, 8, 5, 4, 3, 7, 9,10, 6)$ |
| $ 5, 1, 1, 1, 1, 1 $ | $6048$ | $5$ | $( 5, 7,10, 6, 8)$ |
| $ 5, 2, 2, 1 $ | $90720$ | $10$ | $( 1, 2)( 4, 9)( 5, 6, 7, 8,10)$ |
| $ 5, 4, 1 $ | $181440$ | $20$ | $( 1, 4, 2, 9)( 5, 8, 6,10, 7)$ |
| $ 5, 3, 1, 1 $ | $120960$ | $15$ | $( 1, 3, 4)( 5, 8, 6,10, 7)$ |
| $ 5, 2, 1, 1, 1 $ | $60480$ | $10$ | $( 2, 9)( 5,10, 8, 7, 6)$ |
| $ 5, 3, 2 $ | $120960$ | $30$ | $( 1, 3, 4)( 2, 9)( 5, 8, 6,10, 7)$ |
| $ 4, 2, 2, 2 $ | $18900$ | $4$ | $( 1, 3)( 2, 7)( 4, 6, 5,10)( 8, 9)$ |
| $ 3, 3, 2, 2 $ | $25200$ | $6$ | $( 1, 9, 2)( 3, 8, 7)( 4, 5)( 6,10)$ |
| $ 6, 4 $ | $151200$ | $12$ | $( 1, 7, 9, 3, 2, 8)( 4,10, 5, 6)$ |
| $ 2, 2, 2, 2, 2 $ | $945$ | $2$ | $( 1, 3)( 2, 7)( 4, 5)( 6,10)( 8, 9)$ |
| $ 6, 2, 2 $ | $75600$ | $6$ | $( 1, 8, 2, 3, 9, 7)( 4, 5)( 6,10)$ |
| $ 5, 5 $ | $72576$ | $5$ | $( 1, 2,10, 5, 9)( 3, 7, 6, 4, 8)$ |
| $ 10 $ | $362880$ | $10$ | $( 1, 3, 9, 8, 5, 4,10, 6, 2, 7)$ |
| $ 4, 3, 3 $ | $50400$ | $12$ | $( 1, 2, 9)( 3, 7, 8)( 4,10, 5, 6)$ |
| $ 6, 1, 1, 1, 1 $ | $25200$ | $6$ | $(1,3,2,7,9,8)$ |
| $ 3, 3, 2, 1, 1 $ | $50400$ | $6$ | $( 1, 9, 4)( 2, 7,10)( 6, 8)$ |
| $ 7, 1, 1, 1 $ | $86400$ | $7$ | $( 1, 2, 6, 3, 4,10, 5)$ |
| $ 7, 2, 1 $ | $259200$ | $14$ | $( 1, 4, 2,10, 6, 5, 3)( 7, 9)$ |
| $ 4, 3, 2, 1 $ | $151200$ | $12$ | $( 1, 2, 3, 9)( 4,10)( 6, 7, 8)$ |
| $ 4, 2, 2, 1, 1 $ | $56700$ | $4$ | $( 1, 2,10, 5)( 3, 9)( 4, 6)$ |
| $ 6, 3, 1 $ | $201600$ | $6$ | $( 1, 3, 6, 7,10, 9)( 2, 4, 8)$ |
| $ 4, 3, 1, 1, 1 $ | $50400$ | $12$ | $( 1, 9, 4)( 3, 6, 8, 5)$ |
| $ 7, 3 $ | $172800$ | $21$ | $( 1, 4, 9)( 2, 7, 8,10, 3, 6, 5)$ |
Group invariants
| Order: | $3628800=2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |