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Properties

Label 20T540
Order \(28800\)
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 $t$ :  $540$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,7,11,8,15,10,12)(2,13,4,17,6,18,9,19)(3,20,5,16), (2,8,4,10,5,7)(3,9,6)(11,12,20)(13,17,14)(15,19,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: None

Degree 10: None

Low degree siblings

10T43, 12T288, 20T539, 20T542, 20T544, 24T13996, 24T13997, 24T13998, 25T106, 30T1011, 36T13308, 40T14374, 40T14375, 40T14376, 40T14377, 40T14378, 40T14379, 40T14380, 40T14381

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, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $100$ $2$ $( 2,10)( 4, 7)( 5, 8)(11,20)(13,14)(15,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $225$ $2$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)(11,20)(13,15)(14,19)(17,18)$
$ 3, 3, 3, 3, 3, 3, 1, 1 $ $400$ $3$ $( 1, 2,10)( 3, 7, 4)( 5, 9, 8)(11,20,18)(12,19,15)(13,16,14)$
$ 6, 6, 3, 3, 1, 1 $ $400$ $6$ $( 1, 2,10)( 3, 8, 4, 9, 7, 5)(11,13,18,14,20,16)(12,19,15)$
$ 4, 4, 4, 4, 2, 2 $ $900$ $4$ $( 1, 2, 3, 7)( 4,10)( 5, 9, 6, 8)(11,20,18,17)(12,19,16,14)(13,15)$
$ 5, 5, 5, 5 $ $576$ $5$ $( 1, 2, 3, 6, 8)( 4, 9, 7, 5,10)(11,12,19,16,17)(13,18,14,20,15)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $(11,20)(13,14)(15,19)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $30$ $2$ $(11,20)(13,15)(14,19)(17,18)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $(11,20,18)(12,19,15)(13,16,14)$
$ 6, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $6$ $(11,13,18,14,20,16)(12,19,15)$
$ 4, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $60$ $4$ $(11,20,18,17)(12,19,16,14)(13,15)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $48$ $5$ $(11,12,19,16,17)(13,18,14,20,15)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $300$ $2$ $( 2,10)( 4, 7)( 5, 8)(11,20)(13,15)(14,19)(17,18)$
$ 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1 $ $400$ $6$ $( 2,10)( 4, 7)( 5, 8)(11,20,18)(12,19,15)(13,16,14)$
$ 6, 3, 2, 2, 2, 1, 1, 1, 1, 1 $ $400$ $6$ $( 2,10)( 4, 7)( 5, 8)(11,13,18,14,20,16)(12,19,15)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $600$ $4$ $( 2,10)( 4, 7)( 5, 8)(11,20,18,17)(12,19,16,14)(13,15)$
$ 5, 5, 2, 2, 2, 1, 1, 1, 1 $ $480$ $10$ $( 2,10)( 4, 7)( 5, 8)(11,12,19,16,17)(13,18,14,20,15)$
$ 3, 3, 3, 2, 2, 2, 2, 1, 1, 1 $ $600$ $6$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)(11,20,18)(12,19,15)(13,16,14)$
$ 6, 3, 2, 2, 2, 2, 1, 1, 1 $ $600$ $6$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)(11,13,18,14,20,16)(12,19,15)$
$ 4, 4, 2, 2, 2, 2, 2, 1, 1 $ $900$ $4$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)(11,20,18,17)(12,19,16,14)(13,15)$
$ 5, 5, 2, 2, 2, 2, 1, 1 $ $720$ $10$ $( 2, 7)( 4,10)( 5, 8)( 6, 9)(11,12,19,16,17)(13,18,14,20,15)$
$ 6, 3, 3, 3, 3, 1, 1 $ $800$ $6$ $( 1, 2,10)( 3, 7, 4)( 5, 9, 8)(11,13,18,14,20,16)(12,19,15)$
$ 4, 4, 3, 3, 3, 2, 1 $ $1200$ $12$ $( 1, 2,10)( 3, 7, 4)( 5, 9, 8)(11,20,18,17)(12,19,16,14)(13,15)$
$ 5, 5, 3, 3, 3, 1 $ $960$ $15$ $( 1, 2,10)( 3, 7, 4)( 5, 9, 8)(11,12,19,16,17)(13,18,14,20,15)$
$ 6, 4, 4, 3, 2, 1 $ $1200$ $12$ $( 1, 2,10)( 3, 8, 4, 9, 7, 5)(11,20,18,17)(12,19,16,14)(13,15)$
$ 6, 5, 5, 3, 1 $ $960$ $30$ $( 1, 2,10)( 3, 8, 4, 9, 7, 5)(11,12,19,16,17)(13,18,14,20,15)$
$ 5, 5, 4, 4, 2 $ $1440$ $20$ $( 1, 2, 3, 7)( 4,10)( 5, 9, 6, 8)(11,12,19,16,17)(13,18,14,20,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $120$ $2$ $( 1,12)( 2,19)( 3,16)( 4,13)( 5,20)( 6,17)( 7,14)( 8,11)( 9,18)(10,15)$
$ 4, 4, 4, 2, 2, 2, 2 $ $1200$ $4$ $( 1,12)( 2,15,10,19)( 3,16)( 4,14, 7,13)( 5,11, 8,20)( 6,17)( 9,18)$
$ 4, 4, 4, 4, 2, 2 $ $1800$ $4$ $( 1,12)( 2,14, 7,19)( 3,16)( 4,15,10,13)( 5,11, 8,20)( 6,18, 9,17)$
$ 6, 6, 6, 2 $ $2400$ $6$ $( 1,19, 2,15,10,12)( 3,14, 7,13, 4,16)( 5,18, 9,11, 8,20)( 6,17)$
$ 12, 6, 2 $ $2400$ $12$ $( 1,19, 2,15,10,12)( 3,11, 8,13, 4,18, 9,14, 7,20, 5,16)( 6,17)$
$ 10, 10 $ $2880$ $10$ $( 1,19, 2,16, 3,17, 6,11, 8,12)( 4,18, 9,14, 7,20, 5,15,10,13)$
$ 8, 8, 4 $ $3600$ $8$ $( 1,19, 2,16, 3,14, 7,12)( 4,15,10,13)( 5,18, 9,17, 6,11, 8,20)$

Group invariants

Order:  $28800=2^{7} \cdot 3^{2} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.