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Properties

Label 20T277
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 $t$ :  $277$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,7,12,9,17)(2,19,8,11,10,18), (1,15,17,14,20,2,16,18,13,19)(3,10,12,6,7,4,9,11,5,8)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
60:  $A_5$
120:  $A_5\times C_2$ x 3
240:  20T64
960:  $C_2^4 : A_5$
1920:  $C_2 \wr A_5$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $A_5$

Degree 10: $A_5\times C_2$, $C_2 \wr A_5$ x 2

Low degree siblings

20T277 x 2, 40T2734 x 3, 40T2735 x 3, 40T2765 x 6, 40T2778

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 $ $5$ $2$ $( 7,17)( 8,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3,13)( 4,14)( 7,17)( 8,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3,13)( 4,14)( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 1,12)( 2,11)( 3,13)( 4,14)( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,16)( 6,15)( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3,14)( 4,13)( 5, 6)( 7,18)( 8,17)( 9,19)(10,20)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3,14)( 4,13)( 5, 6)( 7, 8)( 9,19)(10,20)(11,12)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,19)(10,20)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5, 6)( 7,18)( 8,17)( 9,19)(10,20)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,15)( 6,16)( 7,18)( 8,17)( 9,19)(10,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1,19)( 2,20)( 3,14)( 4,13)( 5,18,16, 8)( 6,17,15, 7)( 9,11)(10,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1,19)( 2,20)( 3,14)( 4,13)( 5, 8)( 6, 7)( 9,11)(10,12)(15,17)(16,18)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1,19)( 2,20)( 3, 4)( 5,18,16, 8)( 6,17,15, 7)( 9,11)(10,12)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1,19)( 2,20)( 3, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,14)(15,17)(16,18)$
$ 4, 4, 4, 4, 2, 2 $ $60$ $4$ $( 1,10,12,19)( 2, 9,11,20)( 3,14)( 4,13)( 5,18,16, 8)( 6,17,15, 7)$
$ 4, 4, 4, 4, 2, 2 $ $60$ $4$ $( 1,10,12,19)( 2, 9,11,20)( 3, 4)( 5,18,16, 8)( 6,17,15, 7)(13,14)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $120$ $4$ $( 1, 9,12,20)( 2,10,11,19)( 5, 7)( 6, 8)(15,18)(16,17)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $60$ $4$ $( 1, 9,12,20)( 2,10,11,19)( 5,17,16, 7)( 6,18,15, 8)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1, 9,12,20)( 2,10,11,19)( 3,13)( 4,14)( 5, 7)( 6, 8)(15,18)(16,17)$
$ 4, 4, 4, 4, 2, 2 $ $60$ $4$ $( 1, 9,12,20)( 2,10,11,19)( 3,13)( 4,14)( 5,17,16, 7)( 6,18,15, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 1,20)( 2,19)( 5, 7)( 6, 8)( 9,12)(10,11)(15,18)(16,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1,20)( 2,19)( 3,13)( 4,14)( 5, 7)( 6, 8)( 9,12)(10,11)(15,18)(16,17)$
$ 3, 3, 3, 3, 2, 2, 2, 2 $ $80$ $6$ $( 1,20,16)( 2,19,15)( 3,13)( 4,14)( 5,12, 9)( 6,11,10)( 7,17)( 8,18)$
$ 3, 3, 3, 3, 2, 2, 1, 1, 1, 1 $ $80$ $6$ $( 1,20,16)( 2,19,15)( 3,13)( 4,14)( 5,12, 9)( 6,11,10)$
$ 3, 3, 3, 3, 2, 2, 1, 1, 1, 1 $ $80$ $6$ $( 1,20,16)( 2,19,15)( 5,12, 9)( 6,11,10)( 7,17)( 8,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $80$ $3$ $( 1,20,16)( 2,19,15)( 5,12, 9)( 6,11,10)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1, 9, 5,12,20,16)( 2,10, 6,11,19,15)( 3,13)( 4,14)( 7,17)( 8,18)$
$ 6, 6, 2, 2, 1, 1, 1, 1 $ $80$ $6$ $( 1, 9, 5,12,20,16)( 2,10, 6,11,19,15)( 3,13)( 4,14)$
$ 6, 6, 2, 2, 1, 1, 1, 1 $ $80$ $6$ $( 1, 9, 5,12,20,16)( 2,10, 6,11,19,15)( 7,17)( 8,18)$
$ 6, 6, 1, 1, 1, 1, 1, 1, 1, 1 $ $80$ $6$ $( 1, 9, 5,12,20,16)( 2,10, 6,11,19,15)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,10, 5,11,20,15)( 2, 9, 6,12,19,16)( 3, 4)( 7, 8)(13,14)(17,18)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,10, 5,11,20,15)( 2, 9, 6,12,19,16)( 3, 4)( 7,18)( 8,17)(13,14)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,10, 5,11,20,15)( 2, 9, 6,12,19,16)( 3,14)( 4,13)( 7, 8)(17,18)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,10, 5,11,20,15)( 2, 9, 6,12,19,16)( 3,14)( 4,13)( 7,18)( 8,17)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,19,16, 2,20,15)( 3, 4)( 5,11, 9, 6,12,10)( 7, 8)(13,14)(17,18)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,19,16, 2,20,15)( 3, 4)( 5,11, 9, 6,12,10)( 7,18)( 8,17)(13,14)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,19,16, 2,20,15)( 3,14)( 4,13)( 5,11, 9, 6,12,10)( 7, 8)(17,18)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,19,16, 2,20,15)( 3,14)( 4,13)( 5,11, 9, 6,12,10)( 7,18)( 8,17)$
$ 10, 10 $ $192$ $10$ $( 1,19,16, 8, 3,11, 9, 6,17,14)( 2,20,15, 7, 4,12,10, 5,18,13)$
$ 10, 10 $ $192$ $10$ $( 1,19,16,18,13, 2,20,15,17,14)( 3,11, 9, 6, 7, 4,12,10, 5, 8)$
$ 5, 5, 5, 5 $ $192$ $5$ $( 1, 9, 5, 7,13)( 2,10, 6, 8,14)( 3,12,20,16,17)( 4,11,19,15,18)$
$ 10, 10 $ $192$ $10$ $( 1, 9, 5,17, 3,12,20,16, 7,13)( 2,10, 6,18, 4,11,19,15, 8,14)$
$ 10, 10 $ $192$ $10$ $( 1,19,16, 4, 7,11, 9, 6,13,18)( 2,20,15, 3, 8,12,10, 5,14,17)$
$ 10, 10 $ $192$ $10$ $( 1,19,16, 4,17, 2,20,15, 3,18)( 5,14, 7,11, 9, 6,13, 8,12,10)$
$ 5, 5, 5, 5 $ $192$ $5$ $( 1, 9, 5, 3,17)( 2,10, 6, 4,18)( 7,12,20,16,13)( 8,11,19,15,14)$
$ 10, 10 $ $192$ $10$ $( 1, 9, 5, 3, 7,12,20,16,13,17)( 2,10, 6, 4, 8,11,19,15,14,18)$

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.