↑

Properties

Label 10T19
Order \(200\)
n \(10\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5^2 : C_2$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $19$
Group :  $D_5^2 : C_2$
CHM label :  $[5^{2}:4_{2}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,9,7)(2,4,8,6), (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T21 x 2, 20T48 x 2, 20T50 x 2, 20T55, 20T57 x 2, 25T21, 40T167 x 2, 40T170

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 $ $1$ $1$ $()$
$ 4, 4, 1, 1 $ $50$ $4$ $( 3, 5, 9, 7)( 4, 8,10, 6)$
$ 2, 2, 2, 2, 1, 1 $ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$
$ 5, 1, 1, 1, 1, 1 $ $8$ $5$ $( 2, 4, 6, 8,10)$
$ 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)$
$ 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3, 6)( 4, 7)( 5,10)( 8, 9)$
$ 10 $ $20$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
$ 10 $ $20$ $10$ $( 1, 2, 3, 6, 5,10, 7, 4, 9, 8)$
$ 10 $ $20$ $10$ $( 1, 2, 3, 8, 5, 4, 7,10, 9, 6)$
$ 10 $ $20$ $10$ $( 1, 2, 3,10, 5, 8, 7, 6, 9, 4)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 8, 4,10, 6)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2,10, 8, 6, 4)$

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [200, 43]
Character table:    
      2  3  2  3  .  2  2   1   1   1   1  1  1  1  1
      5  2  .  .  2  1  1   1   1   1   1  2  2  2  2

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

X.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
X.3      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
X.5      2  . -2  2  .  .   .   .   .   .  2  2  2  2
X.6      4  .  . -1 -2  .   A   .   .  *A  B *C  C *B
X.7      4  .  . -1 -2  .  *A   .   .   A *B  C *C  B
X.8      4  .  . -1  . -2   .   A  *A   .  C  B *B *C
X.9      4  .  . -1  . -2   .  *A   A   . *C *B  B  C
X.10     4  .  . -1  .  2   . -*A  -A   . *C *B  B  C
X.11     4  .  . -1  .  2   .  -A -*A   .  C  B *B *C
X.12     4  .  . -1  2  . -*A   .   .  -A *B  C *C  B
X.13     4  .  . -1  2  .  -A   .   . -*A  B *C  C *B
X.14     8  .  .  3  .  .   .   .   .   . -2 -2 -2 -2

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