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Properties

Label 20T285
Order \(3840\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $285$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(2,9)(3,11)(4,12)(5,7,6,8)(13,14)(15,19,16,20)(17,18), (1,12,16,20)(2,11,15,19)(3,17,9,7,4,18,10,8)(5,14)(6,13)
$|\Aut(F/K)|$:  $2$

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 4: None

Degree 5: None

Degree 10: $S_5$

Low degree siblings

10T39 x 2, 20T275, 20T279 x 2, 20T285, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $60$ $4$ $( 1, 2)( 3,12)( 4,11)( 5,19, 6,20)( 7,16, 8,15)( 9,10)(13,17)(14,18)$
$ 4, 4, 4, 2, 2, 2, 1, 1 $ $120$ $4$ $( 1, 2)( 3,11, 4,12)( 5,19, 6,20)( 7,15)( 8,16)(13,17,14,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 3,11)( 4,12)( 5,20)( 6,19)( 7,15)( 8,16)(13,17)(14,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,20)( 6,19)( 7,15)( 8,16)( 9,10)(13,18)(14,17)$
$ 4, 4, 4, 2, 2, 2, 1, 1 $ $120$ $4$ $( 1, 2)( 3,11, 4,12)( 5,20)( 6,19)( 7,16, 8,15)(13,18,14,17)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $4$ $( 3,11)( 4,12)( 5,19, 6,20)( 7,16, 8,15)(13,18)(14,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $60$ $2$ $( 1, 2)( 3,14)( 4,13)( 7,16)( 8,15)( 9,10)(11,17)(12,18)(19,20)$
$ 4, 4, 4, 2, 2, 1, 1, 1, 1 $ $60$ $4$ $( 1, 2)( 3,13, 4,14)( 7,15, 8,16)(11,17,12,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $( 3,13)( 4,14)( 7,16)( 8,15)( 9,10)(11,17)(12,18)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $4$ $( 3,14, 4,13)( 7,15, 8,16)(11,17,12,18)$
$ 4, 4, 4, 2, 2, 1, 1, 1, 1 $ $60$ $4$ $( 1, 2)( 3,13, 4,14)( 7,15, 8,16)( 9,10)(11,18,12,17)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $60$ $2$ $( 1, 2)( 3,14)( 4,13)( 7,16)( 8,15)(11,18)(12,17)$
$ 4, 4, 4, 2, 2, 2, 2 $ $20$ $4$ $( 1, 2)( 3,13, 4,14)( 5, 6)( 7,16, 8,15)( 9,10)(11,18,12,17)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $20$ $2$ $( 1, 2)( 3,14)( 4,13)( 5, 6)( 7,15)( 8,16)(11,18)(12,17)(19,20)$
$ 6, 6, 6, 1, 1 $ $160$ $6$ $( 3, 6,14,18,20,11)( 4, 5,13,17,19,12)( 7,15,10, 8,16, 9)$
$ 12, 6, 2 $ $160$ $12$ $( 1, 2)( 3, 6,13,17,20,11, 4, 5,14,18,19,12)( 7,15,10, 8,16, 9)$
$ 12, 3, 3, 1, 1 $ $160$ $12$ $( 3, 6,13,18,19,12, 4, 5,14,17,20,11)( 7,16, 9)( 8,15,10)$
$ 6, 6, 3, 3, 2 $ $160$ $6$ $( 1, 2)( 3, 6,14,17,19,12)( 4, 5,13,18,20,11)( 7,16, 9)( 8,15,10)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $80$ $3$ $( 3,20,13)( 4,19,14)( 5,11,17)( 6,12,18)( 7,15, 9)( 8,16,10)$
$ 6, 6, 3, 3, 1, 1 $ $80$ $6$ $( 3,20,14, 4,19,13)( 5,12,18, 6,11,17)( 7,16, 9)( 8,15,10)$
$ 6, 3, 3, 3, 3, 2 $ $160$ $6$ $( 1, 2)( 3,19,13, 4,20,14)( 5,11,17)( 6,12,18)( 7,15, 9)( 8,16,10)$
$ 6, 6, 3, 3, 1, 1 $ $160$ $6$ $( 3,19,13)( 4,20,14)( 5,11,18, 6,12,17)( 7,16,10, 8,15, 9)$
$ 6, 6, 6, 2 $ $80$ $6$ $( 1, 2)( 3,20,13, 4,19,14)( 5,11,18, 6,12,17)( 7,16,10, 8,15, 9)$
$ 6, 3, 3, 3, 3, 2 $ $80$ $6$ $( 1, 2)( 3,20,14)( 4,19,13)( 5,12,17)( 6,11,18)( 7,15,10, 8,16, 9)$
$ 8, 4, 4, 2, 2 $ $240$ $8$ $( 1, 9)( 2,10)( 3,13,12,17)( 4,14,11,18)( 5, 8,20,15, 6, 7,19,16)$
$ 4, 4, 4, 4, 4 $ $240$ $4$ $( 1,10, 2, 9)( 3,14,12,17)( 4,13,11,18)( 5, 8,20,16)( 6, 7,19,15)$
$ 8, 4, 4, 2, 2 $ $240$ $8$ $( 1, 9)( 2,10)( 3,14,11,17)( 4,13,12,18)( 5, 8,19,15, 6, 7,20,16)$
$ 4, 4, 4, 4, 4 $ $240$ $4$ $( 1,10, 2, 9)( 3,13,11,17)( 4,14,12,18)( 5, 8,19,16)( 6, 7,20,15)$
$ 5, 5, 5, 5 $ $384$ $5$ $( 1, 9, 6,13, 8)( 2,10, 5,14, 7)( 3,17,11,15,19)( 4,18,12,16,20)$
$ 10, 10 $ $384$ $10$ $( 1, 9, 6,14, 7, 2,10, 5,13, 8)( 3,18,12,15,20, 4,17,11,16,19)$

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.