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Properties

Label 20T140
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 $t$ :  $140$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,9,13,18,11,16,19,3,8)(2,5,10,14,17,12,15,20,4,7), (1,7,16,10,11,17,6,20)(2,8,15,9,12,18,5,19)(3,4)(13,14)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
20:  $F_5$
40:  $F_{5}\times C_2$
320:  $(C_2^4 : C_5):C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $F_5$

Degree 10: $F_5$, $((C_2^4 : C_5):C_4)\times C_2$ x 2

Low degree siblings

10T29 x 2, 20T129, 20T131 x 2, 20T132, 20T133, 20T134, 20T135, 20T137 x 2, 32T34608 x 2, 40T460, 40T462, 40T473, 40T474, 40T475, 40T476, 40T487, 40T488, 40T489, 40T490, 40T557, 40T558 x 2, 40T561, 40T562, 40T563, 40T564, 40T565, 40T566, 40T567 x 2, 40T576, 40T577, 40T578, 40T579, 40T586

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 $ $10$ $2$ $( 5,15)( 6,16)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$
$ 5, 5, 5, 5 $ $64$ $5$ $( 1, 9,18,16, 3)( 2,10,17,15, 4)( 5,14,12,20, 7)( 6,13,11,19, 8)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3,13)( 4,14)( 5,15)( 6,16)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$
$ 10, 10 $ $64$ $10$ $( 1, 9, 8, 6,13,11,19,18,16, 3)( 2,10, 7, 5,14,12,20,17,15, 4)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $20$ $4$ $( 1,19,11, 9)( 2,20,12,10)( 3,18,13, 8)( 4,17,14, 7)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1, 9)( 2,10)( 3,18,13, 8)( 4,17,14, 7)( 5,15)( 6,16)(11,19)(12,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $20$ $2$ $( 1, 9)( 2,10)( 3,18)( 4,17)( 7,14)( 8,13)(11,19)(12,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 9)( 2,10)( 3,18,13, 8)( 4,17,14, 7)(11,19)(12,20)$
$ 4, 4, 4, 4, 2, 2 $ $20$ $4$ $( 1,19,11, 9)( 2,20,12,10)( 3,18,13, 8)( 4,17,14, 7)( 5,15)( 6,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 9)( 2,10)( 3,18)( 4,17)( 5,15)( 6,16)( 7,14)( 8,13)(11,19)(12,20)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1, 7,16,10,11,17, 6,20)( 2, 8,15, 9,12,18, 5,19)( 3, 4)(13,14)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1, 7,16,20)( 2, 8,15,19)( 3,14)( 4,13)( 5, 9,12,18)( 6,10,11,17)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1, 7,16,20)( 2, 8,15,19)( 3, 4)( 5, 9,12,18)( 6,10,11,17)(13,14)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1, 7,16,10,11,17, 6,20)( 2, 8,15, 9,12,18, 5,19)( 3,14)( 4,13)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1, 4,18, 5,11,14, 8,15)( 2, 3,17, 6,12,13, 7,16)( 9,10)(19,20)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1, 4,18,15)( 2, 3,17,16)( 5,11,14, 8)( 6,12,13, 7)( 9,20)(10,19)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1, 4,18, 5,11,14, 8,15)( 2, 3,17, 6,12,13, 7,16)( 9,20)(10,19)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1, 4,18,15)( 2, 3,17,16)( 5,11,14, 8)( 6,12,13, 7)( 9,10)(19,20)$

Group invariants

Order:  $640=2^{7} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [640, 21536]
Character table: Data not available.