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Properties

Label 20T138
Order \(640\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $138$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,2,19)(3,18,4,17)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15), (1,15,17,6,12,4,14,20,7,10,2,16,18,5,11,3,13,19,8,9)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
10:  $D_{5}$
20:  $D_{10}$
40:  $D_{20}$
160:  $(C_2^4 : C_5) : C_2$
320:  $C_2\times (C_2^4 : D_5)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $D_{5}$

Degree 10: $D_{10}$

Low degree siblings

20T138 x 5, 40T456 x 3, 40T478 x 6, 40T479 x 6, 40T537 x 6, 40T543 x 12, 40T547 x 6

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 $ $5$ $2$ $( 1, 2)( 3, 4)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
$ 5, 5, 5, 5 $ $32$ $5$ $( 1,14,18, 7,12)( 2,13,17, 8,11)( 3,15,20, 6, 9)( 4,16,19, 5,10)$
$ 5, 5, 5, 5 $ $32$ $5$ $( 1,18,12,14, 7)( 2,17,11,13, 8)( 3,20, 9,15, 6)( 4,19,10,16, 5)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1,20, 2,19)( 3,18, 4,17)( 5,11)( 6,12)( 7, 9)( 8,10)(13,16)(14,15)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1,20, 2,19)( 3,18, 4,17)( 5,11, 6,12)( 7, 9, 8,10)(13,15)(14,16)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,12, 6,11)( 7,10, 8, 9)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $40$ $2$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,12)( 6,11)( 7,10)( 8, 9)(13,15)(14,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 10, 10 $ $32$ $10$ $( 1,14,18, 8,11, 2,13,17, 7,12)( 3,15,20, 5,10, 4,16,19, 6, 9)$
$ 10, 10 $ $32$ $10$ $( 1,18,11,13, 8, 2,17,12,14, 7)( 3,20,10,16, 5, 4,19, 9,15, 6)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $40$ $2$ $( 3, 4)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13)$
$ 4, 4, 2, 2, 2, 2, 2, 1, 1 $ $40$ $4$ $( 1, 2)( 5,20, 6,19)( 7,18, 8,17)( 9,15)(10,16)(11,14)(12,13)$
$ 4, 4, 4, 4, 2, 1, 1 $ $40$ $4$ $( 3, 4)( 5,19, 6,20)( 7,17, 8,18)( 9,16,10,15)(11,13,12,14)$
$ 4, 4, 2, 2, 2, 2, 2, 1, 1 $ $40$ $4$ $( 1, 2)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16,10,15)(11,13,12,14)$
$ 20 $ $32$ $20$ $( 1,20,12,16, 8, 4,18,10,13, 6, 2,19,11,15, 7, 3,17, 9,14, 5)$
$ 20 $ $32$ $20$ $( 1, 6,13, 9,18, 4, 7,15,12,19, 2, 5,14,10,17, 3, 8,16,11,20)$
$ 20 $ $32$ $20$ $( 1, 9, 7,20,14, 4,12, 5,18,16, 2,10, 8,19,13, 3,11, 6,17,15)$
$ 20 $ $32$ $20$ $( 1,15,18, 5,12, 4,14,19, 8,10, 2,16,17, 6,11, 3,13,20, 7, 9)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,19,18,20)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,19,18,20)$
$ 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,20,18,19)$

Group invariants

Order:  $640=2^{7} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [640, 21459]
Character table: Data not available.