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Properties

Label 20T49
Degree $20$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_5:F_5$

Related objects

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Group action invariants

Degree $n$:  $20$
Transitive number $t$:  $49$
Group:  $C_2\times C_5:F_5$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
Generators:  (1,11,18,7)(2,12,17,8)(3,5,16,13)(4,6,15,14)(9,20,10,19), (1,11,6,7)(2,12,5,8)(3,17,16,9)(4,18,15,10)(13,20,14,19)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$ x 2
$40$:  $F_{5}\times C_2$ x 2
$100$:  $C_5^2 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: $C_5^2 : C_4$

Low degree siblings

20T49, 20T52 x 2, 40T157 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, 2, 2, 2, 1, 1, 1, 1 $ $25$ $2$ $( 5,17)( 6,18)( 7,20)( 8,19)( 9,13)(10,14)(11,16)(12,15)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $( 3, 7,12,15,20)( 4, 8,11,16,19)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $( 3,12,20, 7,15)( 4,11,19, 8,16)$
$ 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 $ $25$ $2$ $( 1, 2)( 3, 4)( 5,18)( 6,17)( 7,19)( 8,20)( 9,14)(10,13)(11,15)(12,16)$
$ 10, 2, 2, 2, 2, 2 $ $4$ $10$ $( 1, 2)( 3, 8,12,16,20, 4, 7,11,15,19)( 5, 6)( 9,10)(13,14)(17,18)$
$ 10, 2, 2, 2, 2, 2 $ $4$ $10$ $( 1, 2)( 3,11,20, 8,15, 4,12,19, 7,16)( 5, 6)( 9,10)(13,14)(17,18)$
$ 4, 4, 4, 4, 4 $ $25$ $4$ $( 1, 3, 2, 4)( 5, 7,18,19)( 6, 8,17,20)( 9,11,14,15)(10,12,13,16)$
$ 4, 4, 4, 4, 4 $ $25$ $4$ $( 1, 3, 2, 4)( 5,20,18, 8)( 6,19,17, 7)( 9,16,14,12)(10,15,13,11)$
$ 4, 4, 4, 4, 4 $ $25$ $4$ $( 1, 4, 2, 3)( 5, 8,18,20)( 6, 7,17,19)( 9,12,14,16)(10,11,13,15)$
$ 4, 4, 4, 4, 4 $ $25$ $4$ $( 1, 4, 2, 3)( 5,19,18, 7)( 6,20,17, 8)( 9,15,14,11)(10,16,13,12)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3, 7,12,15,20)( 4, 8,11,16,19)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3,12,20, 7,15)( 4,11,19, 8,16)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3,15, 7,20,12)( 4,16, 8,19,11)$
$ 10, 10 $ $4$ $10$ $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3, 8,12,16,20, 4, 7,11,15,19)$
$ 10, 10 $ $4$ $10$ $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3,11,20, 8,15, 4,12,19, 7,16)$
$ 10, 10 $ $4$ $10$ $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3,16, 7,19,12, 4,15, 8,20,11)$
$ 10, 10 $ $4$ $10$ $( 1, 9,17, 6,14, 2,10,18, 5,13)( 3,11,20, 8,15, 4,12,19, 7,16)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1,10,17, 5,14)( 2, 9,18, 6,13)( 3,12,20, 7,15)( 4,11,19, 8,16)$

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [200, 48]
Character table:    
      2  3  3  1  1  3  3   1   1  3  3  3  3  1  1  1   1   1   1   1  1
      5  2  .  2  2  2  .   2   2  .  .  .  .  2  2  2   2   2   2   2  2

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

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  C -C  C -C  1  1  1   1   1   1   1  1
X.6      1 -1  1  1  1 -1   1   1 -C  C -C  C  1  1  1   1   1   1   1  1
X.7      1  1  1  1 -1 -1  -1  -1  C  C -C -C  1  1  1  -1  -1  -1  -1  1
X.8      1  1  1  1 -1 -1  -1  -1 -C -C  C  C  1  1  1  -1  -1  -1  -1  1
X.9      4  . -1 -1  4  .  -1  -1  .  .  .  . -1  4 -1  -1   4  -1  -1 -1
X.10     4  . -1 -1 -4  .   1   1  .  .  .  . -1  4 -1   1  -4   1   1 -1
X.11     4  . -1 -1  4  .  -1  -1  .  .  .  . -1 -1  4  -1  -1   4  -1 -1
X.12     4  . -1 -1 -4  .   1   1  .  .  .  . -1 -1  4   1   1  -4   1 -1
X.13     4  .  A *A  4  .   A  *A  .  .  .  .  B -1 -1   B  -1  -1  *B *B
X.14     4  . *A  A  4  .  *A   A  .  .  .  . *B -1 -1  *B  -1  -1   B  B
X.15     4  .  B *B  4  .   B  *B  .  .  .  . *A -1 -1  *A  -1  -1   A  A
X.16     4  . *B  B  4  .  *B   B  .  .  .  .  A -1 -1   A  -1  -1  *A *A
X.17     4  .  A *A -4  .  -A -*A  .  .  .  .  B -1 -1  -B   1   1 -*B *B
X.18     4  . *A  A -4  . -*A  -A  .  .  .  . *B -1 -1 -*B   1   1  -B  B
X.19     4  .  B *B -4  .  -B -*B  .  .  .  . *A -1 -1 -*A   1   1  -A  A
X.20     4  . *B  B -4  . -*B  -B  .  .  .  .  A -1 -1  -A   1   1 -*A *A

A = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5
B = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4
  = (3+Sqrt(5))/2 = 2+b5
C = -E(4)
  = -Sqrt(-1) = -i