Properties

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

Related objects

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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, 45T2246

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 $ $1$ $1$ $()$
$ 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.