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Properties

Label 20T310
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 $t$ :  $310$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,14,5,20,2,9,13,6,19)(3,11,16,7,18,4,12,15,8,17), (5,8,6,7)(9,11,10,12)(13,15,14,16)(17,19)(18,20)
$|\Aut(F/K)|$:  $2$

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: $C_2^4 : C_5$

Low degree siblings

20T310 x 7, 20T314 x 8, 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 $ $5$ $2$ $( 1, 2)( 3, 4)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 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, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4, 2, 3)( 7, 8)( 9,10)(11,12)(17,20)(18,19)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4, 2, 3)( 7, 8)(17,20)(18,19)$
$ 4, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $40$ $4$ $( 1, 4, 2, 3)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,20)(18,19)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4, 2, 3)( 7, 8)(13,14)(15,16)(17,20)(18,19)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $80$ $4$ $( 3, 4)( 7, 8)( 9,12,10,11)(13,14)(15,16)(17,19,18,20)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $80$ $4$ $( 3, 4)( 7, 8)( 9,12,10,11)(17,19,18,20)$
$ 4, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $80$ $4$ $( 1, 4)( 2, 3)( 5, 6)( 9,11)(10,12)(13,16,14,15)(17,19)(18,20)$
$ 4, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $80$ $4$ $( 1, 4)( 2, 3)( 5, 6)( 9,12)(10,11)(13,16,14,15)(17,19)(18,20)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1,14,20, 9, 6)( 2,13,19,10, 5)( 3,16,18,12, 8)( 4,15,17,11, 7)$
$ 10, 10 $ $256$ $10$ $( 1,14,20, 9, 6, 2,13,19,10, 5)( 3,16,18,12, 8, 4,15,17,11, 7)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1,20, 6,14, 9)( 2,19, 5,13,10)( 3,18, 8,16,12)( 4,17, 7,15,11)$
$ 10, 10 $ $256$ $10$ $( 1,20, 6,14, 9, 2,19, 5,13,10)( 3,18, 8,16,12, 4,17, 7,15,11)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1, 6, 9,20,14)( 2, 5,10,19,13)( 3, 8,12,18,16)( 4, 7,11,17,15)$
$ 10, 10 $ $256$ $10$ $( 1, 6, 9,20,14, 2, 5,10,19,13)( 3, 8,12,18,16, 4, 7,11,17,15)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1, 9,14, 6,20)( 2,10,13, 5,19)( 3,12,16, 8,18)( 4,11,15, 7,17)$
$ 10, 10 $ $256$ $10$ $( 1, 9,14, 6,20, 2,10,13, 5,19)( 3,12,16, 8,18, 4,11,15, 7,17)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $2$ $( 1, 2)( 7, 8)(11,12)(15,16)(19,20)$
$ 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4)( 2, 3)( 9,10)(15,16)(17,20,18,19)$
$ 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3)( 2, 4)( 9,10)(15,16)(17,20,18,19)$
$ 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,10)(15,16)(17,20,18,19)$
$ 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(15,16)(17,20,18,19)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 9,12)(10,11)(13,14)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1, 2)( 3, 4)( 9,12)(10,11)(13,14)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $40$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(17,19)(18,20)$
$ 4, 4, 4, 2, 2, 2, 2 $ $80$ $4$ $( 1, 4, 2, 3)( 5, 6)( 7, 8)( 9,11,10,12)(13,16)(14,15)(17,19,18,20)$
$ 4, 4, 4, 2, 2, 1, 1, 1, 1 $ $80$ $4$ $( 1, 4, 2, 3)( 9,11,10,12)(13,16)(14,15)(17,19,18,20)$
$ 10, 5, 5 $ $256$ $10$ $( 1,14,20,10, 5, 2,13,19, 9, 6)( 3,15,18,12, 7)( 4,16,17,11, 8)$
$ 10, 5, 5 $ $256$ $10$ $( 1,14,20,10, 5)( 2,13,19, 9, 6)( 3,15,18,12, 7, 4,16,17,11, 8)$
$ 10, 5, 5 $ $256$ $10$ $( 1,20, 5,13,10, 2,19, 6,14, 9)( 3,17, 7,16,11)( 4,18, 8,15,12)$
$ 10, 5, 5 $ $256$ $10$ $( 1,20, 5,13,10)( 2,19, 6,14, 9)( 3,17, 7,16,11, 4,18, 8,15,12)$
$ 10, 5, 5 $ $256$ $10$ $( 1, 6, 9,20,13, 2, 5,10,19,14)( 3, 7,12,17,15)( 4, 8,11,18,16)$
$ 10, 5, 5 $ $256$ $10$ $( 1, 6, 9,20,13)( 2, 5,10,19,14)( 3, 7,12,17,15, 4, 8,11,18,16)$
$ 10, 5, 5 $ $256$ $10$ $( 1, 9,14, 6,20, 2,10,13, 5,19)( 3,11,16, 7,18)( 4,12,15, 8,17)$
$ 10, 5, 5 $ $256$ $10$ $( 1, 9,14, 6,20)( 2,10,13, 5,19)( 3,11,16, 7,18, 4,12,15, 8,17)$

Group invariants

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