Properties

Label 20T7
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 $t$ :  $7$
Group :  $C_5:D_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,2,12)(3,9,4,10)(5,8,6,7)(13,19,14,20)(15,17,16,18), (1,8,13,20,6,12,18,4,10,16)(2,7,14,19,5,11,17,3,9,15)
$|\Aut(F/K)|$:  $2$

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: $D_{10}$

Low degree siblings

20T11, 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, 2, 2, 2, 2, 1, 1 $ $10$ $2$ $( 3,20)( 4,19)( 5,17)( 6,18)( 7,16)( 8,15)( 9,14)(10,13)(11,12)$
$ 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 $ $10$ $4$ $( 1, 3, 2, 4)( 5,20, 6,19)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$
$ 10, 10 $ $2$ $10$ $( 1, 3, 6, 7,10,11,13,15,18,19)( 2, 4, 5, 8, 9,12,14,16,17,20)$
$ 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 $ $2$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,16)( 6,15)( 7,18)( 8,17)( 9,20)(10,19)$

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

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

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