# Properties

 Label 20T8 Degree $20$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^2\times D_5$

# Related objects

## Group action invariants

 Degree $n$: $20$ Transitive number $t$: $8$ Group: $C_2^2\times D_5$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,19,18,15,13,11,10,7,6,3)(2,20,17,16,14,12,9,8,5,4), (1,20)(2,19)(3,17)(4,18)(5,15)(6,16)(7,14)(8,13)(9,11)(10,12), (1,5,10,14,18,2,6,9,13,17)(3,8,11,16,19,4,7,12,15,20)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$10$:  $D_{5}$
$20$:  $D_{10}$ x 3

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 10: $D_{10}$ x 3

## Low degree siblings

20T8 x 3, 40T10

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $40=2^{3} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [40, 13]
 Character table:  2 3 3 3 3 3 2 3 2 2 2 2 2 2 2 3 3 5 1 . 1 . . 1 . 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 2d 10a 2e 10b 10c 5a 10d 10e 10f 5b 2f 2g 2P 1a 1a 1a 1a 1a 5a 1a 5a 5b 5b 5b 5b 5a 5a 1a 1a 3P 1a 2a 2b 2c 2d 10d 2e 10e 10f 5b 10a 10b 10c 5a 2f 2g 5P 1a 2a 2b 2c 2d 2f 2e 2g 2b 1a 2f 2g 2b 1a 2f 2g 7P 1a 2a 2b 2c 2d 10d 2e 10e 10f 5b 10a 10b 10c 5a 2f 2g X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 X.3 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 X.4 1 -1 1 -1 -1 1 -1 1 1 1 1 1 1 1 1 1 X.5 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 X.6 1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 X.7 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.9 2 . -2 . . A . -A *A -*A *A -*A A -A -2 2 X.10 2 . -2 . . *A . -*A A -A A -A *A -*A -2 2 X.11 2 . -2 . . -*A . *A A -A -A A *A -*A 2 -2 X.12 2 . -2 . . -A . A *A -*A -*A *A A -A 2 -2 X.13 2 . 2 . . A . A -*A -*A *A *A -A -A -2 -2 X.14 2 . 2 . . *A . *A -A -A A A -*A -*A -2 -2 X.15 2 . 2 . . -*A . -*A -A -A -A -A -*A -*A 2 2 X.16 2 . 2 . . -A . -A -*A -*A -*A -*A -A -A 2 2 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5