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Properties

Label 20T314
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 $t$ :  $314$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,15,11,18,2,5,16,12,17)(3,8,13,9,19,4,7,14,10,20), (1,14,19,8,9,4,15,18,6,12)(2,13,20,7,10,3,16,17,5,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $C_{10}$
80:  $C_2^4 : C_5$
160:  $C_2 \times (C_2^4 : C_5)$, 32T2134 x 2
2560:  40T1862

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

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

Low degree siblings

20T310 x 8, 20T314 x 7, 40T3228 x 2, 40T3230 x 2, 40T3241 x 4, 40T3246 x 4, 40T3253 x 4, 40T3255 x 4, 40T3326 x 4, 40T3327 x 4, 40T3410 x 4, 40T3574 x 2, 40T3908 x 2, 40T3936 x 2, 40T3979 x 4, 40T3980 x 4, 40T3984 x 4, 40T3995 x 4, 40T4003 x 4, 40T4009 x 4, 40T4013 x 4, 40T4021 x 4, 40T4022 x 4, 40T4027 x 4, 40T4041 x 8, 40T4168 x 2, 40T4297 x 4, 40T4299 x 4, 40T4304 x 4, 40T4315 x 4, 40T4317 x 4, 40T4328 x 4, 40T4517 x 4, 40T4532 x 4, 40T4704 x 4, 40T4705 x 4, 40T4717 x 4

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $(13,14)(15,16)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 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, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $40$ $4$ $( 1, 2)( 3, 4)( 5, 8)( 6, 7)( 9,10)(11,12)(13,15,14,16)(19,20)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 2)( 3, 4)( 5, 8)( 6, 7)(13,15,14,16)(19,20)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 5, 8)( 6, 7)( 9,10)(11,12)(13,15,14,16)(19,20)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 5, 8)( 6, 7)(13,15,14,16)(19,20)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $80$ $4$ $( 1, 2)( 5, 7)( 6, 8)( 9,12,10,11)(15,16)(19,20)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $80$ $4$ $( 1, 2)( 5, 7)( 6, 8)( 9,12,10,11)(15,16)(17,18)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $80$ $4$ $( 5, 8, 6, 7)( 9,12)(10,11)(13,15,14,16)(17,19)(18,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $80$ $4$ $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,12)(10,11)(13,15,14,16)(17,19)(18,20)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1,15,18, 5,12)( 2,16,17, 6,11)( 3,13,19, 7,10)( 4,14,20, 8, 9)$
$ 10, 10 $ $256$ $10$ $( 1,16,17, 6,11, 2,15,18, 5,12)( 3,14,20, 8, 9, 4,13,19, 7,10)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1,18,12,15, 5)( 2,17,11,16, 6)( 3,19,10,13, 7)( 4,20, 9,14, 8)$
$ 10, 10 $ $256$ $10$ $( 1,18,12,16, 6, 2,17,11,15, 5)( 3,19,10,14, 8, 4,20, 9,13, 7)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1,12, 5,18,15)( 2,11, 6,17,16)( 3,10, 7,19,13)( 4, 9, 8,20,14)$
$ 10, 10 $ $256$ $10$ $( 1,12, 5,18,16, 2,11, 6,17,15)( 3,10, 7,19,14, 4, 9, 8,20,13)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1, 5,15,12,18)( 2, 6,16,11,17)( 3, 7,13,10,19)( 4, 8,14, 9,20)$
$ 10, 10 $ $256$ $10$ $( 1, 5,16,11,17, 2, 6,15,12,18)( 3, 7,14, 9,20, 4, 8,13,10,19)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1, 4)( 2, 3)( 7, 8)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1, 4)( 2, 3)( 7, 8)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1, 4)( 2, 3)( 7, 8)( 9,10)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $2$ $( 1, 4)( 2, 3)( 7, 8)( 9,10)(11,12)(15,16)(17,18)(19,20)$
$ 4, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $40$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,10)(11,12)(13,15)(14,16)(19,20)$
$ 4, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $40$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,10)(11,12)(13,16)(14,15)(19,20)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)(13,15)(14,16)(19,20)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)(13,16)(14,15)(19,20)$
$ 4, 4, 4, 2, 1, 1, 1, 1, 1, 1 $ $80$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(19,20)$
$ 4, 4, 4, 2, 2, 2, 1, 1 $ $80$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,14)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)$
$ 10, 10 $ $256$ $10$ $( 1,14,20, 8,10, 3,16,18, 5,12)( 2,13,19, 7, 9, 4,15,17, 6,11)$
$ 10, 10 $ $256$ $10$ $( 1,13,19, 7, 9, 4,16,18, 5,12)( 2,14,20, 8,10, 3,15,17, 6,11)$
$ 10, 10 $ $256$ $10$ $( 1,20, 9,14, 8, 3,17,11,16, 5)( 2,19,10,13, 7, 4,18,12,15, 6)$
$ 10, 10 $ $256$ $10$ $( 1,20, 9,13, 7, 4,18,12,16, 5)( 2,19,10,14, 8, 3,17,11,15, 6)$
$ 10, 10 $ $256$ $10$ $( 1, 9, 8,19,13, 3,11, 6,17,16)( 2,10, 7,20,14, 4,12, 5,18,15)$
$ 10, 10 $ $256$ $10$ $( 1, 9, 8,19,14, 4,12, 5,18,16)( 2,10, 7,20,13, 3,11, 6,17,15)$
$ 10, 10 $ $256$ $10$ $( 1, 8,13,10,19, 3, 6,16,12,18)( 2, 7,14, 9,20, 4, 5,15,11,17)$
$ 10, 10 $ $256$ $10$ $( 1, 8,14, 9,20, 4, 5,16,12,18)( 2, 7,13,10,19, 3, 6,15,11,17)$

Group invariants

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