# Properties

 Label 20T21 Degree $20$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_4\times D_5$

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## Group action invariants

 Degree $n$: $20$ Transitive number $t$: $21$ Group: $D_4\times D_5$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,13,7,20,3,15,9,11,6,17)(2,14,8,19,4,16,10,12,5,18), (1,14,2,13)(3,12,4,11)(5,20,6,19)(7,18,8,17)(9,16,10,15), (1,11,4,14,6,15,8,18,9,20,2,12,3,13,5,16,7,17,10,19)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$10$:  $D_{5}$
$16$:  $D_4\times C_2$
$20$:  $D_{10}$ x 3
$40$:  20T8

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 10: $D_{10}$

## Low degree siblings

20T21 x 3, 40T22 x 2, 40T39 x 2, 40T40 x 2

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

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 39]
 Character table:  2 4 3 3 4 4 4 3 2 3 2 3 3 2 2 3 3 2 2 3 3 5 1 1 . . 1 . 1 1 1 1 1 1 1 1 . . 1 1 1 1 1a 2a 2b 2c 2d 2e 5a 10a 10b 10c 10d 5b 10e 20a 2f 4a 20b 10f 2g 4b 2P 1a 1a 1a 1a 1a 1a 5b 5b 5b 5a 5a 5a 5a 10b 1a 2d 10d 5b 1a 2d 3P 1a 2a 2b 2c 2d 2e 5b 10c 10d 10a 10b 5a 10f 20b 2f 4a 20a 10e 2g 4b 5P 1a 2a 2b 2c 2d 2e 1a 2a 2d 2a 2d 1a 2g 4b 2f 4a 4b 2g 2g 4b 7P 1a 2a 2b 2c 2d 2e 5b 10c 10d 10a 10b 5a 10f 20b 2f 4a 20a 10e 2g 4b 11P 1a 2a 2b 2c 2d 2e 5a 10a 10b 10c 10d 5b 10e 20a 2f 4a 20b 10f 2g 4b 13P 1a 2a 2b 2c 2d 2e 5b 10c 10d 10a 10b 5a 10f 20b 2f 4a 20a 10e 2g 4b 17P 1a 2a 2b 2c 2d 2e 5b 10c 10d 10a 10b 5a 10f 20b 2f 4a 20a 10e 2g 4b 19P 1a 2a 2b 2c 2d 2e 5a 10a 10b 10c 10d 5b 10e 20a 2f 4a 20b 10f 2g 4b X.1 1 1 1 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 1 -1 -1 1 X.3 1 -1 -1 1 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 1 -1 -1 1 X.5 1 -1 1 -1 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 -1 -1 -1 -1 X.7 1 1 -1 -1 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 -1 -1 -1 -1 X.9 2 . . -2 -2 2 2 . -2 . -2 2 . . . . . . . . X.10 2 . . 2 -2 -2 2 . -2 . -2 2 . . . . . . . . X.11 2 -2 . . 2 . A -A A -*A *A *A -*A *A . . A -A -2 2 X.12 2 -2 . . 2 . *A -*A *A -A A A -A A . . *A -*A -2 2 X.13 2 -2 . . 2 . A -A A -*A *A *A *A -*A . . -A A 2 -2 X.14 2 -2 . . 2 . *A -*A *A -A A A A -A . . -*A *A 2 -2 X.15 2 2 . . 2 . A A A *A *A *A -*A -*A . . -A -A -2 -2 X.16 2 2 . . 2 . *A *A *A A A A -A -A . . -*A -*A -2 -2 X.17 2 2 . . 2 . A A A *A *A *A *A *A . . A A 2 2 X.18 2 2 . . 2 . *A *A *A A A A A A . . *A *A 2 2 X.19 4 . . . -4 . B . -B . -*B *B . . . . . . . . X.20 4 . . . -4 . *B . -*B . -B B . . . . . . . . A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5