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Properties

Label 20T11
Order \(40\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:D_4$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $11$
Group :  $C_5:D_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,2,18)(3,16,4,15)(5,14,6,13)(7,12,8,11)(9,20,10,19), (1,4,6,7,10)(2,3,5,8,9)(11,13,16,17,20,12,14,15,18,19)
$|\Aut(F/K)|$:  $10$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 10: $D_5$

Low degree siblings

20T7, 40T11

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

Group invariants

Order:  $40=2^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [40, 8]
Character table:    
      2  3  2  3   2   2   2  2   2   2   2  2  2  2
      5  1  1  1   1   1   1  1   1   1   1  1  .  .

        1a 2a 2b 10a 10b 10c 5a 10d 10e 10f 5b 2c 4a
     2P 1a 1a 1a  5b  5b  5b 5b  5a  5a  5a 5a 1a 2b
     3P 1a 2a 2b 10d 10f 10e 5b 10a 10b 10c 5a 2c 4a
     5P 1a 2a 2b  2b  2a  2a 1a  2b  2a  2a 1a 2c 4a
     7P 1a 2a 2b 10d 10e 10f 5b 10a 10c 10b 5a 2c 4a

X.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
X.3      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
X.5      2  . -2  -2   .   .  2  -2   .   .  2  .  .
X.6      2 -2  2   A  -A  -A  A  *A -*A -*A *A  .  .
X.7      2 -2  2  *A -*A -*A *A   A  -A  -A  A  .  .
X.8      2  2  2   A   A   A  A  *A  *A  *A *A  .  .
X.9      2  2  2  *A  *A  *A *A   A   A   A  A  .  .
X.10     2  . -2 -*A   B  -B *A  -A   C  -C  A  .  .
X.11     2  . -2 -*A  -B   B *A  -A  -C   C  A  .  .
X.12     2  . -2  -A   C  -C  A -*A  -B   B *A  .  .
X.13     2  . -2  -A  -C   C  A -*A   B  -B *A  .  .

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
B = -E(5)+E(5)^4
C = -E(5)^2+E(5)^3