↑

Properties

Label 20T75
Order \(320\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times C_2^4:C_5$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $75$
Group :  $C_4\times C_2^4:C_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,7,19,14,4,11,6,18,16,2,10,8,20,13,3,12,5,17,15), (1,16,17,5,11,3,14,20,8,9,2,15,18,6,12,4,13,19,7,10)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
5:  $C_5$
10:  $C_{10}$
20:  20T1
80:  $C_2^4 : C_5$
160:  $C_2 \times (C_2^4 : C_5)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_{10}$

Low degree siblings

20T75 x 2, 40T198 x 3, 40T289 x 3, 40T290 x 6, 40T291 x 12

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

Group invariants

Order:  $320=2^{6} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [320, 1584]
Character table: Data not available.