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Properties

Label 20T375
Order \(7680\)
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 $t$ :  $375$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,5,15,12,4,19,7,14,9,2,17,6,16,11,3,20,8,13,10), (1,12)(2,11)(3,9)(4,10)(7,8)(13,14)(17,18), (1,12)(2,11)(3,9)(4,10)(5,20,14,6,19,13)(7,17,16)(8,18,15)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
120:  $S_5$
240:  $S_5\times C_2$
480:  20T120
1920:  $(C_2^4:A_5) : C_2$
3840:  $C_2 \wr S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $S_5$

Degree 10: $S_5\times C_2$

Low degree siblings

20T375, 40T5473 x 2, 40T5476 x 2, 40T5505, 40T5521 x 2, 40T5522 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $(17,18)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 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, 4, 4, 4, 4 $ $20$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)$
$ 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,20,18,19)$
$ 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,20,18,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $120$ $2$ $( 1,18)( 2,17)( 3,19)( 4,20)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1,17, 2,18)( 3,20, 4,19)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $40$ $2$ $( 1,18)( 2,17)( 3,19)( 4,20)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1,17, 2,18)( 3,20, 4,19)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$
$ 4, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $120$ $4$ $( 1,20, 2,19)( 3,18, 4,17)( 5, 6)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $120$ $2$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5, 6)(11,12)(15,16)$
$ 4, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1,20, 2,19)( 3,18, 4,17)( 5, 6)(11,12)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5, 6)(11,12)(13,14)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $120$ $4$ $( 1,19, 2,20)( 3,18, 4,17)( 5,12)( 6,11)( 7, 9)( 8,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5,12)( 6,11)( 7, 9)( 8,10)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1,19, 2,20)( 3,18, 4,17)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $60$ $2$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(15,16)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $60$ $4$ $( 1,19, 2,20)( 3,18, 4,17)( 5,11, 6,12)( 7,10, 8, 9)$
$ 4, 4, 4, 4, 2, 2 $ $60$ $4$ $( 1,19, 2,20)( 3,18, 4,17)( 5,11, 6,12)( 7,10, 8, 9)(13,14)(15,16)$
$ 4, 4, 4, 2, 2, 2, 2 $ $240$ $4$ $( 1,18, 2,17)( 3,20, 4,19)( 5, 9)( 6,10)( 7,11)( 8,12)(13,15,14,16)$
$ 4, 2, 2, 2, 2, 2, 2, 2, 2 $ $120$ $4$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5, 9)( 6,10)( 7,11)( 8,12)(13,15,14,16)$
$ 4, 4, 4, 4, 4 $ $120$ $4$ $( 1,18, 2,17)( 3,20, 4,19)( 5,10, 6, 9)( 7,12, 8,11)(13,15,14,16)$
$ 3, 3, 3, 3, 2, 2, 2, 2 $ $80$ $6$ $( 1,20,12)( 2,19,11)( 3,17,10)( 4,18, 9)( 5, 6)( 7, 8)(13,14)(15,16)$
$ 6, 6, 2, 2, 2, 2 $ $80$ $6$ $( 1,19,11, 2,20,12)( 3,18, 9, 4,17,10)( 5, 6)( 7, 8)(13,14)(15,16)$
$ 3, 3, 3, 3, 2, 2, 1, 1, 1, 1 $ $160$ $6$ $( 1,20,12)( 2,19,11)( 3,17,10)( 4,18, 9)( 5, 6)( 7, 8)$
$ 6, 6, 2, 2, 1, 1, 1, 1 $ $160$ $6$ $( 1,19,11, 2,20,12)( 3,18, 9, 4,17,10)( 5, 6)( 7, 8)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $80$ $3$ $( 1,20,12)( 2,19,11)( 3,17,10)( 4,18, 9)$
$ 6, 6, 1, 1, 1, 1, 1, 1, 1, 1 $ $80$ $6$ $( 1,19,11, 2,20,12)( 3,18, 9, 4,17,10)$
$ 12, 4, 4 $ $160$ $12$ $( 1,17,12, 4,20, 9, 2,18,11, 3,19,10)( 5, 7, 6, 8)(13,16,14,15)$
$ 12, 4, 4 $ $160$ $12$ $( 1,17,12, 4,20, 9, 2,18,11, 3,19,10)( 5, 7, 6, 8)(13,15,14,16)$
$ 12, 4, 4 $ $160$ $12$ $( 1,18,11, 3,20, 9, 2,17,12, 4,19,10)( 5, 7, 6, 8)(13,15,14,16)$
$ 12, 4, 4 $ $160$ $12$ $( 1,17,12, 4,20, 9, 2,18,11, 3,19,10)( 5, 8, 6, 7)(13,15,14,16)$
$ 6, 6, 4, 4 $ $320$ $12$ $( 1,18,11, 3,19, 9)( 2,17,12, 4,20,10)( 5,15, 6,16)( 7,13, 8,14)$
$ 6, 6, 2, 2, 2, 2 $ $320$ $6$ $( 1,18,11, 3,19, 9)( 2,17,12, 4,20,10)( 5,16)( 6,15)( 7,14)( 8,13)$
$ 6, 3, 3, 2, 2, 2, 2 $ $320$ $6$ $( 1,20,12, 2,19,11)( 3,18,10)( 4,17, 9)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 6, 4, 4, 3, 3 $ $320$ $12$ $( 1,20,12, 2,19,11)( 3,18,10)( 4,17, 9)( 5,13, 6,14)( 7,16, 8,15)$
$ 8, 8, 2, 1, 1 $ $480$ $8$ $( 1,19,11, 6, 2,20,12, 5)( 3,17, 9, 8, 4,18,10, 7)(13,14)$
$ 4, 4, 4, 4, 2, 1, 1 $ $480$ $4$ $( 1,20,12, 5)( 2,19,11, 6)( 3,18,10, 7)( 4,17, 9, 8)(13,14)$
$ 8, 8, 2, 2 $ $480$ $8$ $( 1,18,12, 8, 2,17,11, 7)( 3,19,10, 5, 4,20, 9, 6)(13,16)(14,15)$
$ 4, 4, 4, 4, 2, 2 $ $480$ $4$ $( 1,17,11, 7)( 2,18,12, 8)( 3,20, 9, 6)( 4,19,10, 5)(13,16)(14,15)$
$ 20 $ $384$ $20$ $( 1,18,11, 7,14, 3,20, 9, 5,15, 2,17,12, 8,13, 4,19,10, 6,16)$
$ 20 $ $384$ $20$ $( 1,17,12, 8,13, 4,20, 9, 5,15, 2,18,11, 7,14, 3,19,10, 6,16)$
$ 5, 5, 5, 5 $ $384$ $5$ $( 1,20,11, 5,14)( 2,19,12, 6,13)( 3,17, 9, 8,15)( 4,18,10, 7,16)$
$ 10, 10 $ $384$ $10$ $( 1,19,12, 6,13, 2,20,11, 5,14)( 3,18,10, 7,16, 4,17, 9, 8,15)$

Group invariants

Order:  $7680=2^{9} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.