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Properties

Label 20T324
Order \(5120\)
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 $t$ :  $324$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,3,17)(2,19,4,18)(5,11,8,9,6,12,7,10)(15,16), (1,6,9,19,16,2,5,10,20,15)(3,7,12,17,13,4,8,11,18,14), (1,13,19,9,7)(2,14,20,10,8)(3,15,18,11,5)(4,16,17,12,6)
$|\Aut(F/K)|$:  $2$

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 3, 20T325 x 4, 20T354 x 4, 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 $ $5$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5, 6)( 7, 8)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(17,20)(18,19)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,14)(15,16)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1, 3)( 2, 4)( 7, 8)(13,15)(14,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $2$ $( 1, 3)( 2, 4)( 5, 6)( 9,10)(11,12)(13,15)(14,16)(19,20)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3, 2, 4)( 7, 8)(11,12)(17,19,18,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3, 2, 4)( 7, 8)(11,12)(13,14)(15,16)(17,20,18,19)$
$ 5, 5, 5, 5 $ $512$ $5$ $( 1, 8,10,19,15)( 2, 7, 9,20,16)( 3, 5,11,18,13)( 4, 6,12,17,14)$
$ 5, 5, 5, 5 $ $512$ $5$ $( 1,10,15, 8,19)( 2, 9,16, 7,20)( 3,11,13, 5,18)( 4,12,14, 6,17)$
$ 8, 4, 4, 2, 1, 1 $ $160$ $8$ $( 1, 9, 4,11)( 2,10, 3,12)( 7, 8)(13,18,16,19,14,17,15,20)$
$ 8, 4, 4, 2, 1, 1 $ $160$ $8$ $( 1,10, 3,11)( 2, 9, 4,12)( 5, 6)(13,18,16,19,14,17,15,20)$
$ 8, 4, 2, 2, 2, 2 $ $160$ $8$ $( 1,12, 4, 9, 2,11, 3,10)( 5, 8)( 6, 7)(13,19,14,20)(15,17)(16,18)$
$ 8, 4, 2, 2, 2, 2 $ $160$ $8$ $( 1,11, 3, 9, 2,12, 4,10)( 5, 7)( 6, 8)(13,19,14,20)(15,17)(16,18)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $80$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)(13,18,14,17)(15,19,16,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $80$ $4$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5, 6)( 7, 8)(13,18,14,17)(15,19,16,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $80$ $2$ $( 1,10)( 2, 9)( 3,11)( 4,12)(13,17)(14,18)(15,20)(16,19)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $80$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5, 6)( 7, 8)(13,17)(14,18)(15,20)(16,19)$
$ 4, 4, 4, 4, 2, 2 $ $160$ $4$ $( 1,12, 2,11)( 3,10)( 4, 9)( 5, 8, 6, 7)(13,19,16,17)(14,20,15,18)$
$ 4, 4, 4, 4, 2, 2 $ $160$ $4$ $( 1,11)( 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,19,16,17)(14,20,15,18)$
$ 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 $ $5$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 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, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(17,19)(18,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,14)(15,16)(17,20)(18,19)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1, 3)( 2, 4)( 7, 8)(13,15)(14,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $2$ $( 1, 3)( 2, 4)( 5, 6)( 9,10)(11,12)(13,15)(14,16)(17,18)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3, 2, 4)( 7, 8)(11,12)(17,20,18,19)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3, 2, 4)( 7, 8)(11,12)(13,14)(15,16)(17,19,18,20)$
$ 10, 10 $ $512$ $10$ $( 1, 8,10,19,16, 2, 7, 9,20,15)( 3, 5,11,18,14, 4, 6,12,17,13)$
$ 10, 10 $ $512$ $10$ $( 1,10,15, 8,19, 2, 9,16, 7,20)( 3,11,13, 5,18, 4,12,14, 6,17)$
$ 8, 4, 4, 2, 1, 1 $ $160$ $8$ $( 1, 9, 4,11)( 2,10, 3,12)( 7, 8)(13,18,15,20,14,17,16,19)$
$ 8, 4, 4, 2, 1, 1 $ $160$ $8$ $( 1,10, 3,11)( 2, 9, 4,12)( 5, 6)(13,18,15,20,14,17,16,19)$
$ 8, 4, 2, 2, 2, 2 $ $160$ $8$ $( 1,12, 4, 9, 2,11, 3,10)( 5, 8)( 6, 7)(13,19)(14,20)(15,17,16,18)$
$ 8, 4, 2, 2, 2, 2 $ $160$ $8$ $( 1,11, 3, 9, 2,12, 4,10)( 5, 7)( 6, 8)(13,19)(14,20)(15,17,16,18)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $80$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)(13,18)(14,17)(15,19)(16,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $80$ $2$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5, 6)( 7, 8)(13,18)(14,17)(15,19)(16,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $80$ $4$ $( 1,10)( 2, 9)( 3,11)( 4,12)(13,17,14,18)(15,20,16,19)$
$ 4, 4, 4, 4, 2, 2 $ $80$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5, 6)( 7, 8)(13,17,14,18)(15,20,16,19)$
$ 4, 4, 4, 4, 2, 2 $ $160$ $4$ $( 1,12, 2,11)( 3,10)( 4, 9)( 5, 8, 6, 7)(13,19,15,18)(14,20,16,17)$
$ 4, 4, 4, 4, 2, 2 $ $160$ $4$ $( 1,11)( 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,19,15,18)(14,20,16,17)$

Group invariants

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