Properties

 Label 10T45 Order $$3628800$$ n $$10$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $S_{10}$

Related objects

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 $\leq 47$

Degree 2: None

Degree 5: None

Low degree siblings

20T1007, 45T2246

Siblings 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1, 1, 1, 1, 1, 1, 1$ $45$ $2$ $( 1, 8)$ $7, 1, 1, 1$ $86400$ $7$ $( 2, 9, 3, 5,10, 6, 7)$ $7, 2, 1$ $259200$ $14$ $( 1, 8)( 2,10, 9, 6, 3, 7, 5)$ $2, 2, 1, 1, 1, 1, 1, 1$ $630$ $2$ $( 3, 9)( 4, 5)$ $3, 3, 1, 1, 1, 1$ $8400$ $3$ $( 1, 2,10)( 6, 8, 7)$ $4, 1, 1, 1, 1, 1, 1$ $1260$ $4$ $( 3, 5, 9, 4)$ $3, 3, 2, 2$ $25200$ $6$ $( 1,10, 2)( 3, 9)( 4, 5)( 6, 7, 8)$ $4, 3, 3$ $50400$ $12$ $( 1, 2,10)( 3, 4, 9, 5)( 6, 8, 7)$ $4, 2, 2, 2$ $18900$ $4$ $( 1, 7)( 2, 6)( 3, 4, 9, 5)( 8,10)$ $2, 2, 2, 2, 1, 1$ $4725$ $2$ $( 1, 7)( 2, 6)( 3, 9)( 4, 5)$ $4, 4, 1, 1$ $56700$ $4$ $( 1, 2, 7, 6)( 3, 5, 9, 4)$ $4, 4, 2$ $56700$ $4$ $( 1, 2, 7, 6)( 3, 5, 9, 4)( 8,10)$ $4, 2, 2, 1, 1$ $56700$ $4$ $( 1, 7)( 2, 6)( 3, 4, 9, 5)$ $4, 2, 1, 1, 1, 1$ $18900$ $4$ $( 2, 6)( 3, 5, 9, 4)$ $2, 2, 2, 1, 1, 1, 1$ $3150$ $2$ $( 1, 7)( 2, 6)( 8,10)$ $6, 1, 1, 1, 1$ $25200$ $6$ $( 1, 8, 2, 7,10, 6)$ $3, 3, 2, 1, 1$ $50400$ $6$ $( 1, 2,10)( 3, 5)( 6, 8, 7)$ $8, 2$ $226800$ $8$ $( 1, 6, 9, 4, 7, 2, 3, 5)( 8,10)$ $3, 1, 1, 1, 1, 1, 1, 1$ $240$ $3$ $( 3,10, 5)$ $3, 2, 2, 1, 1, 1$ $25200$ $6$ $( 1, 7)( 2, 6)( 3, 5,10)$ $4, 3, 2, 1$ $151200$ $12$ $( 1, 6, 7, 2)( 3,10, 5)( 4, 8)$ $4, 3, 1, 1, 1$ $50400$ $12$ $( 1,10, 6)( 3, 4, 9, 5)$ $3, 3, 3, 1$ $22400$ $3$ $( 1, 2, 7)( 3, 5,10)( 6, 9, 8)$ $6, 3, 1$ $201600$ $6$ $( 1, 8, 2, 6, 7, 9)( 3,10, 5)$ $3, 2, 2, 2, 1$ $25200$ $6$ $( 1, 6)( 2, 9)( 3, 5,10)( 7, 8)$ $6, 2, 1, 1$ $151200$ $6$ $( 1, 9, 5, 7, 3, 4)( 8,10)$ $5, 1, 1, 1, 1, 1$ $6048$ $5$ $( 2, 3, 6, 7, 9)$ $5, 3, 1, 1$ $120960$ $15$ $( 1, 8, 4)( 2, 6, 9, 3, 7)$ $5, 2, 1, 1, 1$ $60480$ $10$ $( 2, 9, 7, 6, 3)( 5,10)$ $8, 1, 1$ $226800$ $8$ $( 1, 9, 4, 6, 7, 3, 5, 2)$ $2, 2, 2, 2, 2$ $945$ $2$ $( 1, 7)( 2, 6)( 3, 9)( 4, 5)( 8,10)$ $6, 2, 2$ $75600$ $6$ $( 1, 9, 5, 7, 3, 4)( 2, 6)( 8,10)$ $5, 2, 2, 1$ $90720$ $10$ $( 1, 2, 8,10, 6)( 3, 9)( 4, 5)$ $5, 4, 1$ $181440$ $20$ $( 1,10, 2, 6, 8)( 3, 4, 9, 5)$ $3, 2, 1, 1, 1, 1, 1$ $5040$ $6$ $( 1, 4, 8)( 5,10)$ $5, 3, 2$ $120960$ $30$ $( 1, 8, 4)( 2, 7, 3, 9, 6)( 5,10)$ $5, 5$ $72576$ $5$ $( 1, 9, 7, 2, 3)( 4, 8, 5,10, 6)$ $10$ $362880$ $10$ $( 1, 4, 7, 5, 3, 6, 9, 8, 2,10)$ $6, 4$ $151200$ $12$ $( 1, 8, 4, 5, 2, 9)( 3,10, 6, 7)$ $9, 1$ $403200$ $9$ $( 1, 6, 5, 7, 8, 3, 2, 9,10)$ $7, 3$ $172800$ $21$ $( 1,10, 8, 4, 6, 2, 5)( 3, 9, 7)$

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.