↑

Properties

Label 20T354
Order \(5120\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $354$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,12)(3,10,4,9)(5,6)(7,8)(13,20,15,18,14,19,16,17), (1,13)(2,14)(3,16,4,15)(5,18,7,19)(6,17,8,20)(11,12), (1,2)(7,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
10:  $D_{5}$
20:  $D_{10}$
160:  $(C_2^4 : C_5) : C_2$
320:  $C_2\times (C_2^4 : D_5)$
2560:  20T241

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $D_{5}$

Degree 10: $(C_2^4 : C_5) : C_2$

Low degree siblings

20T324 x 4, 20T325 x 4, 20T354 x 3, 40T3447 x 4, 40T3448 x 4, 40T3449 x 4, 40T3455 x 8, 40T3457 x 8, 40T3490, 40T3494 x 4, 40T3498 x 2, 40T3499 x 2, 40T4365 x 2, 40T4373 x 2, 40T4377 x 2, 40T4378 x 2, 40T4549 x 2, 40T4550 x 2, 40T4555 x 4, 40T4727 x 8, 40T4755 x 4, 40T4762 x 4, 40T4765 x 8, 40T4823 x 8, 40T4933 x 4, 40T4938 x 4, 40T4940 x 8, 40T4989 x 4, 40T4991 x 4, 40T4992 x 4, 40T4994 x 4, 40T5000 x 8, 40T5002 x 8

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 $ $5$ $2$ $(13,14)(15,16)(17,18)(19,20)$
$ 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 $ $5$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 3, 4)( 9,10)(13,15,14,16)(17,20,18,19)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,15,14,16)(17,20,18,19)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4)( 2, 3)( 5, 8, 6, 7)(13,15)(14,16)(17,20,18,19)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1, 4)( 2, 3)( 5, 7, 6, 8)( 9,10)(11,12)(13,15)(14,16)(17,20,18,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $2$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1, 4)( 2, 3)( 9,11)(10,12)(15,16)(17,18)$
$ 5, 5, 5, 5 $ $512$ $5$ $( 1,12,14, 5,20)( 2,11,13, 6,19)( 3, 9,16, 7,17)( 4,10,15, 8,18)$
$ 5, 5, 5, 5 $ $512$ $5$ $( 1,14,20,12, 5)( 2,13,19,11, 6)( 3,16,17, 9, 7)( 4,15,18,10, 8)$
$ 4, 4, 4, 2, 2, 2, 1, 1 $ $160$ $4$ $( 1,20, 4,18)( 2,19, 3,17)( 5,11)( 6,12)( 7,10, 8, 9)(15,16)$
$ 4, 4, 4, 2, 2, 2, 1, 1 $ $160$ $4$ $( 1,19, 3,18)( 2,20, 4,17)( 5,11)( 6,12)( 7,10, 8, 9)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $80$ $2$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,11)( 6,12)( 7, 9)( 8,10)(13,15)(14,16)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $80$ $4$ $( 1,17, 2,18)( 3,19, 4,20)( 5,11)( 6,12)( 7, 9)( 8,10)(13,16)(14,15)$
$ 4, 4, 4, 4, 2, 2 $ $80$ $4$ $( 1,18, 2,17)( 3,20, 4,19)( 5,11, 6,12)( 7, 9, 8,10)(13,15)(14,16)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $80$ $4$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,11, 6,12)( 7, 9, 8,10)(13,16)(14,15)$
$ 8, 4, 2, 2, 1, 1, 1, 1 $ $160$ $8$ $( 1,20)( 2,19)( 3,17, 4,18)( 5,11, 8,10, 6,12, 7, 9)$
$ 8, 4, 2, 2, 2, 2 $ $160$ $8$ $( 1,19, 2,20)( 3,18)( 4,17)( 5,11, 8,10, 6,12, 7, 9)(13,14)(15,16)$
$ 8, 4, 4, 4 $ $160$ $8$ $( 1,18, 4,19, 2,17, 3,20)( 5,11, 8, 9)( 6,12, 7,10)(13,15,14,16)$
$ 8, 4, 4, 4 $ $160$ $8$ $( 1,17, 3,19, 2,18, 4,20)( 5,11, 8, 9)( 6,12, 7,10)(13,16,14,15)$
$ 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 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(17,18)(19,20)$
$ 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 3, 4)( 9,10)(13,15,14,16)(17,19,18,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,15,14,16)(17,19,18,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4)( 2, 3)( 5, 8, 6, 7)(13,15)(14,16)(17,19,18,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1, 4)( 2, 3)( 5, 7, 6, 8)( 9,10)(11,12)(13,15)(14,16)(17,19,18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $2$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1, 4)( 2, 3)( 9,11)(10,12)(15,16)(19,20)$
$ 10, 10 $ $512$ $10$ $( 1,12,14, 5,20, 2,11,13, 6,19)( 3, 9,16, 7,17, 4,10,15, 8,18)$
$ 10, 10 $ $512$ $10$ $( 1,14,20,11, 6, 2,13,19,12, 5)( 3,16,17,10, 8, 4,15,18, 9, 7)$
$ 4, 4, 4, 2, 2, 2, 1, 1 $ $160$ $4$ $( 1,20, 3,17)( 2,19, 4,18)( 5,11)( 6,12)( 7,10, 8, 9)(15,16)$
$ 4, 4, 4, 2, 2, 2, 1, 1 $ $160$ $4$ $( 1,19, 4,17)( 2,20, 3,18)( 5,11)( 6,12)( 7,10, 8, 9)(13,14)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $80$ $4$ $( 1,18, 2,17)( 3,20, 4,19)( 5,11)( 6,12)( 7, 9)( 8,10)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $80$ $2$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,11)( 6,12)( 7, 9)( 8,10)(13,16)(14,15)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $80$ $4$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,11, 6,12)( 7, 9, 8,10)(13,15)(14,16)$
$ 4, 4, 4, 4, 2, 2 $ $80$ $4$ $( 1,17, 2,18)( 3,19, 4,20)( 5,11, 6,12)( 7, 9, 8,10)(13,16)(14,15)$
$ 8, 4, 2, 2, 1, 1, 1, 1 $ $160$ $8$ $( 1,20, 2,19)( 3,17)( 4,18)( 5,11, 8,10, 6,12, 7, 9)$
$ 8, 4, 2, 2, 2, 2 $ $160$ $8$ $( 1,19)( 2,20)( 3,18, 4,17)( 5,11, 8,10, 6,12, 7, 9)(13,14)(15,16)$
$ 8, 4, 4, 4 $ $160$ $8$ $( 1,18, 3,20, 2,17, 4,19)( 5,11, 8, 9)( 6,12, 7,10)(13,15,14,16)$
$ 8, 4, 4, 4 $ $160$ $8$ $( 1,17, 4,20, 2,18, 3,19)( 5,11, 8, 9)( 6,12, 7,10)(13,16,14,15)$

Group invariants

Order:  $5120=2^{10} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.