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Properties

Label 20T135
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 $t$ :  $135$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,2,3)(5,13,20,11)(6,14,19,12)(7,15,18,10)(8,16,17,9), (1,20,2,19)(3,17,4,18)(5,16)(6,15)(7,13)(8,14)(9,10)(11,12)
$|\Aut(F/K)|$:  $4$

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: $F_5$

Low degree siblings

10T29 x 2, 20T129, 20T131 x 2, 20T132, 20T133, 20T134, 20T137 x 2, 20T140, 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 $ $10$ $2$ $( 1, 2)( 3, 4)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(19,20)$
$ 5, 5, 5, 5 $ $64$ $5$ $( 1,20,16,10, 6)( 2,19,15, 9, 5)( 3,17,14,11, 7)( 4,18,13,12, 8)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 5,20)( 6,19)( 7,18)( 8,17)( 9,15,10,16)(11,14,12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 2)( 3, 4)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16)(10,15)(11,13)(12,14)$
$ 4, 4, 4, 4, 2, 2 $ $20$ $4$ $( 1, 2)( 3, 4)( 5,19, 6,20)( 7,17, 8,18)( 9,15,10,16)(11,14,12,13)$
$ 4, 4, 4, 4, 4 $ $40$ $4$ $( 1, 4, 2, 3)( 5,13,20,11)( 6,14,19,12)( 7,15,18,10)( 8,16,17, 9)$
$ 8, 8, 4 $ $40$ $8$ $( 1, 3, 2, 4)( 5,14,19,12, 6,13,20,11)( 7,16,17, 9, 8,15,18,10)$
$ 8, 8, 4 $ $40$ $8$ $( 1, 4, 2, 3)( 5,11,20,14, 6,12,19,13)( 7,10,18,16, 8, 9,17,15)$
$ 4, 4, 4, 4, 4 $ $40$ $4$ $( 1, 3, 2, 4)( 5,11,20,13)( 6,12,19,14)( 7,10,18,15)( 8, 9,17,16)$
$ 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 $ $10$ $2$ $( 1, 2)( 3, 4)(13,14)(15,16)(17,18)(19,20)$
$ 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 $ $64$ $10$ $( 1,20,15, 9, 5, 2,19,16,10, 6)( 3,17,13,12, 8, 4,18,14,11, 7)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $20$ $4$ $( 5,20, 6,19)( 7,18, 8,17)( 9,15,10,16)(11,14,12,13)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1, 2)( 3, 4)( 5,20, 6,19)( 7,18, 8,17)( 9,16)(10,15)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $20$ $2$ $( 5,19)( 6,20)( 7,17)( 8,18)( 9,16)(10,15)(11,13)(12,14)$
$ 8, 8, 4 $ $40$ $8$ $( 1, 4, 2, 3)( 5,13,20,12, 6,14,19,11)( 7,15,18, 9, 8,16,17,10)$
$ 4, 4, 4, 4, 4 $ $40$ $4$ $( 1, 3, 2, 4)( 5,14,19,11)( 6,13,20,12)( 7,16,17,10)( 8,15,18, 9)$
$ 4, 4, 4, 4, 4 $ $40$ $4$ $( 1, 4, 2, 3)( 5,11,20,13)( 6,12,19,14)( 7,10,18,15)( 8, 9,17,16)$
$ 8, 8, 4 $ $40$ $8$ $( 1, 3, 2, 4)( 5,11,20,14, 6,12,19,13)( 7,10,18,16, 8, 9,17,15)$

Group invariants

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