Properties

Label 20T21
Order \(80\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_4\times D_5$

Related objects

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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)
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)
$|\Aut(F/K)|$:  $2$

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 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, 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