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Properties

Label 20T291
Order \(3840\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $291$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,13,3,2,20,14,4)(5,16,9,17)(6,15,10,18)(7,12)(8,11), (1,15)(2,16)(3,18,4,17)(5,12,6,11)(7,8)(13,20,14,19)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
120:  $S_5$
1920:  16T1329

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 10: $S_5$

Low degree siblings

12T257 x 2, 20T281 x 2, 20T291, 24T7238, 24T7255, 32T206828 x 4, 40T2727 x 2, 40T2744, 40T2745, 40T2755 x 2, 40T2837 x 2, 40T2844 x 2

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 $ $15$ $2$ $( 9,10)(11,12)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(19,20)$
$ 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, 4, 2, 2, 2, 2, 2, 1, 1 $ $120$ $4$ $( 3, 9, 4,10)( 5,14)( 6,13)( 7,17, 8,18)(11,12)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 3,10)( 4, 9)( 5,14)( 6,13)( 7,18)( 8,17)(15,16)$
$ 4, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $120$ $4$ $( 3, 9, 4,10)( 5,13, 6,14)( 7,18)( 8,17)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $40$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5,14)( 6,13)( 7,17)( 8,18)(11,12)(19,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $4$ $( 3, 8)( 4, 7)( 9,20,10,19)(11,17,12,18)(13,15)(14,16)$
$ 4, 4, 4, 2, 2, 2, 1, 1 $ $240$ $4$ $( 3, 7, 4, 8)( 5, 6)( 9,20,10,19)(11,18)(12,17)(13,15,14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 3, 7)( 4, 8)( 9,19)(10,20)(11,17)(12,18)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,19)(10,20)(11,17)(12,18)(13,16)(14,15)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $60$ $4$ $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,20,10,19)(11,17,12,18)(13,16)(14,15)$
$ 6, 6, 6, 1, 1 $ $320$ $6$ $( 3,20, 9, 7,12,17)( 4,19,10, 8,11,18)( 5,16,13, 6,15,14)$
$ 6, 6, 3, 3, 2 $ $320$ $6$ $( 1, 2)( 3,19,10, 7,11,17)( 4,20, 9, 8,12,18)( 5,15,14)( 6,16,13)$
$ 6, 6, 6, 2 $ $320$ $6$ $( 1, 2)( 3,12, 9, 4,11,10)( 5,15,13, 6,16,14)( 7,19,17, 8,20,18)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $320$ $3$ $( 3,11,10)( 4,12, 9)( 5,16,14)( 6,15,13)( 7,20,17)( 8,19,18)$
$ 5, 5, 5, 5 $ $384$ $5$ $( 1,20,14,16,10)( 2,19,13,15, 9)( 3,17,11, 7, 5)( 4,18,12, 8, 6)$
$ 10, 10 $ $384$ $10$ $( 1,20,13,15, 9, 2,19,14,16,10)( 3,18,12, 7, 6, 4,17,11, 8, 5)$
$ 8, 4, 4, 2, 2 $ $240$ $8$ $( 1,20,13, 4, 2,19,14, 3)( 5,16, 9,17)( 6,15,10,18)( 7,12)( 8,11)$
$ 4, 4, 4, 4, 4 $ $240$ $4$ $( 1,19,14, 3)( 2,20,13, 4)( 5,16,10,17)( 6,15, 9,18)( 7,11, 8,12)$
$ 8, 4, 4, 2, 2 $ $240$ $8$ $( 1,19,14, 4, 2,20,13, 3)( 5,16, 9,17)( 6,15,10,18)( 7,11)( 8,12)$
$ 4, 4, 4, 4, 4 $ $240$ $4$ $( 1,20,13, 3)( 2,19,14, 4)( 5,16,10,17)( 6,15, 9,18)( 7,12, 8,11)$

Group invariants

Order:  $3840=2^{8} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.