Properties

Label 25T38
Order \(500\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5.D_5^2$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $38$
Group :  $C_5.D_5^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,2,5,3)(6,25,10,21,9,22,8,23,7,24)(11,18,14,20,12,17,15,19,13,16), (1,14,10,23,20,4,15,7,25,16)(2,11,9,22,17,3,13,8,21,19)(5,12,6,24,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
10:  $D_{5}$ x 2
20:  $D_{10}$ x 2
100:  $D_5^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Low degree siblings

25T38

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, 1, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ $10$ $5$ $( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ $10$ $5$ $( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $25$ $2$ $( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18)(15,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19) (15,20)$
$ 10, 10, 2, 2, 1 $ $50$ $10$ $( 2, 5)( 3, 4)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$
$ 10, 10, 2, 2, 1 $ $50$ $10$ $( 2, 5)( 3, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,18,15,17,14,16,13,20,12,19)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6,10, 9, 8, 7)(11,14,12,15,13)(16,18,20,17,19) (21,22,23,24,25)$
$ 10, 10, 5 $ $50$ $10$ $( 1, 2, 3, 4, 5)( 6,21, 9,23, 7,25,10,22, 8,24)(11,17,12,16,13,20,14,19,15,18)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 9, 7,10, 8)(11,12,13,14,15)(16,20,19,18,17) (21,23,25,22,24)$
$ 10, 10, 5 $ $50$ $10$ $( 1, 3, 5, 2, 4)( 6,21, 7,25, 8,24, 9,23,10,22)(11,18,13,16,15,19,12,17,14,20)$
$ 10, 10, 5 $ $50$ $10$ $( 1, 6,16,11,21)( 2, 7,18,13,22, 5,10,19,14,25)( 3, 8,20,15,23, 4, 9,17,12,24)$
$ 5, 5, 5, 5, 5 $ $10$ $5$ $( 1, 6,16,11,21)( 2,10,18,14,22)( 3, 9,20,12,23)( 4, 8,17,15,24) ( 5, 7,19,13,25)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,17,13,24)( 2,10,19,11,25)( 3, 9,16,14,21)( 4, 8,18,12,22) ( 5, 7,20,15,23)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,18,15,22)( 2,10,20,13,23)( 3, 9,17,11,24)( 4, 8,19,14,25) ( 5, 7,16,12,21)$
$ 10, 10, 5 $ $50$ $10$ $( 1,11, 6,21,16)( 2,13,10,25,18, 5,14, 7,22,19)( 3,15, 9,24,20, 4,12, 8,23,17)$
$ 5, 5, 5, 5, 5 $ $10$ $5$ $( 1,11, 6,21,16)( 2,14,10,22,18)( 3,12, 9,23,20)( 4,15, 8,24,17) ( 5,13, 7,25,19)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1,11, 8,25,17)( 2,14, 7,21,19)( 3,12, 6,22,16)( 4,15,10,23,18) ( 5,13, 9,24,20)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1,11,10,24,18)( 2,14, 9,25,20)( 3,12, 8,21,17)( 4,15, 7,22,19) ( 5,13, 6,23,16)$

Group invariants

Order:  $500=2^{2} \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [500, 27]
Character table:   
      2  2  1  1  2  2  2   1   1  1   1  1   1   1  1  .  .   1  1  .  .
      5  3  2  2  1  1  1   1   1  3   1  3   1   1  2  2  2   1  2  2  2

        1a 5a 5b 2a 2b 2c 10a 10b 5c 10c 5d 10d 10e 5e 5f 5g 10f 5h 5i 5j
     2P 1a 5b 5a 1a 1a 1a  5b  5a 5d  5d 5c  5c  5h 5h 5i 5j  5e 5e 5f 5g
     3P 1a 5b 5a 2a 2b 2c 10b 10a 5d 10d 5c 10c 10f 5h 5i 5j 10e 5e 5f 5g
     5P 1a 1a 1a 2a 2b 2c  2c  2c 1a  2a 1a  2a  2b 1a 1a 1a  2b 1a 1a 1a
     7P 1a 5b 5a 2a 2b 2c 10b 10a 5d 10d 5c 10c 10f 5h 5i 5j 10e 5e 5f 5g

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      2  2  2  . -2  .   .   .  2   .  2   . -*A *A *A *A  -A  A  A  A
X.6      2  2  2  . -2  .   .   .  2   .  2   .  -A  A  A  A -*A *A *A *A
X.7      2  2  2  .  2  .   .   .  2   .  2   .   A  A  A  A  *A *A *A *A
X.8      2  2  2  .  2  .   .   .  2   .  2   .  *A *A *A *A   A  A  A  A
X.9      2  A *A  .  . -2  -A -*A  2   .  2   .   .  2 *A  A   .  2  A *A
X.10     2 *A  A  .  . -2 -*A  -A  2   .  2   .   .  2  A *A   .  2 *A  A
X.11     2  A *A  .  .  2   A  *A  2   .  2   .   .  2 *A  A   .  2  A *A
X.12     2 *A  A  .  .  2  *A   A  2   .  2   .   .  2  A *A   .  2 *A  A
X.13     4  B *B  .  .  .   .   .  4   .  4   .   .  B -1 *D   . *B -1  D
X.14     4 *B  B  .  .  .   .   .  4   .  4   .   . *B -1  D   .  B -1 *D
X.15     4  B *B  .  .  .   .   .  4   .  4   .   . *B  D -1   .  B *D -1
X.16     4 *B  B  .  .  .   .   .  4   .  4   .   .  B *D -1   . *B  D -1
X.17    10  .  . -2  .  .   .   .  C  -A *C -*A   .  .  .  .   .  .  .  .
X.18    10  .  . -2  .  .   .   . *C -*A  C  -A   .  .  .  .   .  .  .  .
X.19    10  .  .  2  .  .   .   .  C   A *C  *A   .  .  .  .   .  .  .  .
X.20    10  .  .  2  .  .   .   . *C  *A  C   A   .  .  .  .   .  .  .  .

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