Properties

Label 25T39
Order \(500\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:C_{10}.C_2$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $39$
Group :  $C_5^2:C_{10}.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,12,21)(2,18,11,24)(3,20,15,22)(4,17,14,25)(5,19,13,23)(6,7,9,8), (1,15,21,8,20)(2,11,22,9,16)(3,12,23,10,17)(4,13,24,6,18)(5,14,25,7,19)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$
20:  $F_5$, 20T2
100:  20T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Low degree siblings

25T36

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 $ $10$ $5$ $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ $10$ $5$ $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)(21,24,22,25,23)$
$ 4, 4, 4, 4, 4, 4, 1 $ $125$ $4$ $( 2, 3, 5, 4)( 6,16,21,11)( 7,18,25,14)( 8,20,24,12)( 9,17,23,15)(10,19,22,13)$
$ 4, 4, 4, 4, 4, 4, 1 $ $125$ $4$ $( 2, 4, 5, 3)( 6,11,21,16)( 7,14,25,18)( 8,12,24,20)( 9,15,23,17)(10,13,22,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 10, 10, 2, 2, 1 $ $50$ $10$ $( 2, 5)( 3, 4)( 6,22, 9,24, 7,21,10,23, 8,25)(11,18,12,17,13,16,14,20,15,19)$
$ 10, 10, 2, 2, 1 $ $50$ $10$ $( 2, 5)( 3, 4)( 6,23, 7,22, 8,21, 9,25,10,24)(11,20,13,18,15,16,12,19,14,17)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,11,19,23)( 2, 7,12,20,24)( 3, 8,13,16,25)( 4, 9,14,17,21) ( 5,10,15,18,22)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,12,17,24)( 2, 7,13,18,25)( 3, 8,14,19,21)( 4, 9,15,20,22) ( 5,10,11,16,23)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,13,20,25)( 2, 7,14,16,21)( 3, 8,15,17,22)( 4, 9,11,18,23) ( 5,10,12,19,24)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,14,18,21)( 2, 7,15,19,22)( 3, 8,11,20,23)( 4, 9,12,16,24) ( 5,10,13,17,25)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,15,16,22)( 2, 7,11,17,23)( 3, 8,12,18,24)( 4, 9,13,19,25) ( 5,10,14,20,21)$

Group invariants

Order:  $500=2^{2} \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [500, 21]
Character table:   
      2  2  1  1  2  2  2   1   1  .  .  .  .  .  .
      5  3  2  2  .  .  1   1   1  3  2  2  2  2  2

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

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  C -C -1  -1  -1  1  1  1  1  1  1
X.4      1  1  1 -C  C -1  -1  -1  1  1  1  1  1  1
X.5      2  A *A  .  . -2  -A -*A  2 *A *A  A  2  A
X.6      2 *A  A  .  . -2 -*A  -A  2  A  A *A  2 *A
X.7      2  A *A  .  .  2   A  *A  2 *A *A  A  2  A
X.8      2 *A  A  .  .  2  *A   A  2  A  A *A  2 *A
X.9      4  4  4  .  .  .   .   .  4 -1 -1 -1 -1 -1
X.10     4  B *B  .  .  .   .   .  4  D /D  E -1 /E
X.11     4  B *B  .  .  .   .   .  4 /D  D /E -1  E
X.12     4 *B  B  .  .  .   .   .  4  E /E /D -1  D
X.13     4 *B  B  .  .  .   .   .  4 /E  E  D -1 /D
X.14    20  .  .  .  .  .   .   . -5  .  .  .  .  .

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 = -E(4)
  = -Sqrt(-1) = -i
D = -E(5)-E(5)^2+E(5)^3
E = -E(5)-E(5)^3+E(5)^4