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Properties

Label 20T6
Degree $20$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4\times D_5$

Related objects

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Group action invariants

Degree $n$:  $20$
Transitive number $t$:  $6$
Group:  $C_4\times D_5$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $4$
Generators:  (1,15,2,16)(3,14,4,13)(5,12,6,11)(7,9,8,10)(17,20,18,19), (1,3,5,8,10,11,14,16,18,19,2,4,6,7,9,12,13,15,17,20)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$10$:  $D_{5}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $D_{5}$

Degree 10: $D_{10}$

Low degree siblings

20T6, 40T9

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

Group invariants

Order:  $40=2^{3} \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [40, 5]
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 4a 20a 4b 20b 10a  5a 20c 20d 10b  5b 4c 4d
     2P 1a 1a 1a 1a 2b 10a 2b 10a  5b  5b 10b 10b  5a  5a 2b 2b
     3P 1a 2a 2b 2c 4b 20d 4a 20c 10b  5b 20b 20a 10a  5a 4d 4c
     5P 1a 2a 2b 2c 4a  4c 4b  4d  2b  1a  4c  4d  2b  1a 4c 4d
     7P 1a 2a 2b 2c 4b 20d 4a 20c 10b  5b 20b 20a 10a  5a 4d 4c
    11P 1a 2a 2b 2c 4b 20b 4a 20a 10a  5a 20d 20c 10b  5b 4d 4c
    13P 1a 2a 2b 2c 4a 20c 4b 20d 10b  5b 20a 20b 10a  5a 4c 4d
    17P 1a 2a 2b 2c 4a 20c 4b 20d 10b  5b 20a 20b 10a  5a 4c 4d
    19P 1a 2a 2b 2c 4b 20b 4a 20a 10a  5a 20d 20c 10b  5b 4d 4c

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  A  -A -A   A  -1   1  -A   A  -1   1 -A  A
X.6      1 -1 -1  1 -A   A  A  -A  -1   1   A  -A  -1   1  A -A
X.7      1  1 -1 -1  A   A -A  -A  -1   1   A  -A  -1   1  A -A
X.8      1  1 -1 -1 -A  -A  A   A  -1   1  -A   A  -1   1 -A  A
X.9      2  . -2  .  .   B  .  -B  *D -*D   C  -C   D  -D  E -E
X.10     2  . -2  .  .   C  .  -C   D  -D   B  -B  *D -*D  E -E
X.11     2  . -2  .  .  -C  .   C   D  -D  -B   B  *D -*D -E  E
X.12     2  . -2  .  .  -B  .   B  *D -*D  -C   C   D  -D -E  E
X.13     2  .  2  .  .   D  .   D -*D -*D  *D  *D  -D  -D -2 -2
X.14     2  .  2  .  .  *D  .  *D  -D  -D   D   D -*D -*D -2 -2
X.15     2  .  2  .  . -*D  . -*D  -D  -D  -D  -D -*D -*D  2  2
X.16     2  .  2  .  .  -D  .  -D -*D -*D -*D -*D  -D  -D  2  2

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(20)-E(20)^9
C = -E(20)^13-E(20)^17
D = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
E = -2*E(4)
  = -2*Sqrt(-1) = -2i