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Properties

Label 10T39
Order \(3840\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_2 \wr S_5$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $39$
Group :  $C_2 \wr S_5$
CHM label :  $[2^{5}]S(5)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (5,10), (2,10)(5,7), (1,3,5,7,9)(2,4,6,8,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
120:  $S_5$
240:  $S_5\times C_2$
1920:  $(C_2^4:A_5) : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $S_5$

Low degree siblings

10T39, 20T275, 20T279 x 2, 20T285 x 2, 20T288 x 2, 20T289 x 2, 30T517 x 2, 30T524 x 2, 32T206825 x 2, 40T2728 x 2, 40T2731 x 2, 40T2748, 40T2749, 40T2757 x 2, 40T2771 x 2, 40T2772 x 2, 40T2773 x 2, 40T2774 x 2, 40T2779 x 2, 40T2780 x 2, 40T2781 x 2, 40T2782 x 2, 40T2798, 40T2839 x 2

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 $ $5$ $2$ $( 5,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 2, 7)( 5,10)$
$ 2, 2, 2, 1, 1, 1, 1 $ $10$ $2$ $( 2, 7)( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 1, 1 $ $5$ $2$ $( 1, 6)( 2, 7)( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $( 4,10)( 5, 9)$
$ 4, 1, 1, 1, 1, 1, 1 $ $20$ $4$ $( 4, 5, 9,10)$
$ 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 2, 7)( 4,10)( 5, 9)$
$ 4, 2, 1, 1, 1, 1 $ $60$ $4$ $( 2, 7)( 4, 5, 9,10)$
$ 2, 2, 2, 2, 1, 1 $ $60$ $2$ $( 1, 6)( 2, 7)( 4,10)( 5, 9)$
$ 4, 2, 2, 1, 1 $ $60$ $4$ $( 1, 6)( 2, 7)( 4, 5, 9,10)$
$ 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4,10)( 5, 9)$
$ 4, 2, 2, 2 $ $20$ $4$ $( 1, 6)( 2, 7)( 3, 8)( 4, 5, 9,10)$
$ 3, 3, 1, 1, 1, 1 $ $80$ $3$ $( 3, 9, 5)( 4,10, 8)$
$ 6, 1, 1, 1, 1 $ $80$ $6$ $( 3, 9,10, 8, 4, 5)$
$ 3, 3, 2, 1, 1 $ $160$ $6$ $( 2, 7)( 3, 9, 5)( 4,10, 8)$
$ 6, 2, 1, 1 $ $160$ $6$ $( 2, 7)( 3, 9,10, 8, 4, 5)$
$ 3, 3, 2, 2 $ $80$ $6$ $( 1, 6)( 2, 7)( 3, 9, 5)( 4,10, 8)$
$ 6, 2, 2 $ $80$ $6$ $( 1, 6)( 2, 7)( 3, 9,10, 8, 4, 5)$
$ 2, 2, 2, 2, 1, 1 $ $60$ $2$ $( 2, 8)( 3, 7)( 4,10)( 5, 9)$
$ 4, 2, 2, 1, 1 $ $120$ $4$ $( 2, 8)( 3, 7)( 4, 5, 9,10)$
$ 4, 4, 1, 1 $ $60$ $4$ $( 2, 8, 7, 3)( 4, 5, 9,10)$
$ 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 6)( 2, 8)( 3, 7)( 4,10)( 5, 9)$
$ 4, 2, 2, 2 $ $120$ $4$ $( 1, 6)( 2, 8)( 3, 7)( 4, 5, 9,10)$
$ 4, 4, 2 $ $60$ $4$ $( 1, 6)( 2, 8, 7, 3)( 4, 5, 9,10)$
$ 4, 4, 1, 1 $ $240$ $4$ $( 2, 8, 4,10)( 3, 9, 5, 7)$
$ 8, 1, 1 $ $240$ $8$ $( 2, 8, 4, 5, 7, 3, 9,10)$
$ 4, 4, 2 $ $240$ $4$ $( 1, 6)( 2, 8, 4,10)( 3, 9, 5, 7)$
$ 8, 2 $ $240$ $8$ $( 1, 6)( 2, 8, 4, 5, 7, 3, 9,10)$
$ 3, 3, 2, 2 $ $160$ $6$ $( 1, 7)( 2, 6)( 3, 9, 5)( 4,10, 8)$
$ 6, 2, 2 $ $160$ $6$ $( 1, 7)( 2, 6)( 3, 9,10, 8, 4, 5)$
$ 4, 3, 3 $ $160$ $12$ $( 1, 2, 6, 7)( 3, 9, 5)( 4,10, 8)$
$ 6, 4 $ $160$ $12$ $( 1, 2, 6, 7)( 3, 9,10, 8, 4, 5)$
$ 5, 5 $ $384$ $5$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$
$ 10 $ $384$ $10$ $( 1, 7, 3, 9,10, 6, 2, 8, 4, 5)$

Group invariants

Order:  $3840=2^{8} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.