Properties

Label 25T44
Order \(600\)
n \(25\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $44$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,6,5,16,10)(2,19,7,4,17,9)(3,18,8)(11,23)(12,22)(13,21)(14,25)(15,24), (1,24,2,25,3,21,4,22,5,23)(6,9,7,10,8)(11,16,12,17,13,18,14,19,15,20)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
10:  $D_{5}$
120:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$, $S_5$

Low degree siblings

30T144

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, 5 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $50$ $2$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,16)( 6,10)( 7, 9)(12,15)(13,14)(21,25)(22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1,16)( 2,17)( 3,18)( 4,19)( 5,20)( 6,21)( 7,22)( 8,23)( 9,24)(10,25)$
$ 10, 10, 5 $ $30$ $10$ $( 1,17, 3,19, 5,16, 2,18, 4,20)( 6,22, 8,24,10,21, 7,23, 9,25)(11,12,13,14,15)$
$ 10, 10, 5 $ $30$ $10$ $( 1,18, 5,17, 4,16, 3,20, 2,19)( 6,23,10,22, 9,21, 8,25, 7,24)(11,13,15,12,14)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1,16,21)( 2,17,22)( 3,18,23)( 4,19,24)( 5,20,25)$
$ 15, 5, 5 $ $40$ $15$ $( 1,17,23, 4,20,21, 2,18,24, 5,16,22, 3,19,25)( 6, 7, 8, 9,10)(11,12,13,14,15)$
$ 15, 5, 5 $ $40$ $15$ $( 1,18,25, 2,19,21, 3,20,22, 4,16,23, 5,17,24)( 6, 8,10, 7, 9)(11,13,15,12,14)$
$ 6, 6, 3, 2, 2, 2, 2, 2 $ $100$ $6$ $( 1,20,21, 5,16,25)( 2,19,22, 4,17,24)( 3,18,23)( 6,13)( 7,12)( 8,11)( 9,15) (10,14)$
$ 4, 4, 4, 4, 4, 2, 2, 1 $ $150$ $4$ $( 1,20,21,10)( 2,19,22, 9)( 3,18,23, 8)( 4,17,24, 7)( 5,16,25, 6)(12,15) (13,14)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,16,21, 6,14)( 2,17,22, 7,15)( 3,18,23, 8,11)( 4,19,24, 9,12) ( 5,20,25,10,13)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,17,23, 9,13)( 2,18,24,10,14)( 3,19,25, 6,15)( 4,20,21, 7,11) ( 5,16,22, 8,12)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,18,25, 7,12)( 2,19,21, 8,13)( 3,20,22, 9,14)( 4,16,23,10,15) ( 5,17,24, 6,11)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,20,24, 8,15)( 2,16,25, 9,11)( 3,17,21,10,12)( 4,18,22, 6,13) ( 5,19,23, 7,14)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1,19,22,10,11)( 2,20,23, 6,12)( 3,16,24, 7,13)( 4,17,25, 8,14) ( 5,18,21, 9,15)$

Group invariants

Order:  $600=2^{3} \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [600, 145]
Character table:   
      2  3  2  2  2  3   2   2  1   .   .  1  2  .   .   .   .   .
      3  1  1  1  1  .   .   .  1   1   1  1  .  .   .   .   .   .
      5  2  2  2  .  1   1   1  1   1   1  .  .  2   2   2   2   2

        1a 5a 5b 2a 2b 10a 10b 3a 15a 15b 6a 4a 5c  5d  5e  5f  5g
     2P 1a 5b 5a 1a 1a  5b  5a 3a 15b 15a 3a 2b 5c  5g  5d  5e  5f
     3P 1a 5b 5a 2a 2b 10b 10a 1a  5b  5a 2a 4a 5c  5e  5f  5g  5d
     5P 1a 1a 1a 2a 2b  2b  2b 3a  3a  3a 6a 4a 1a  1a  1a  1a  1a
     7P 1a 5b 5a 2a 2b 10b 10a 3a 15b 15a 6a 4a 5c  5g  5d  5e  5f
    11P 1a 5a 5b 2a 2b 10a 10b 3a 15a 15b 6a 4a 5c  5d  5e  5f  5g
    13P 1a 5b 5a 2a 2b 10b 10a 3a 15b 15a 6a 4a 5c  5e  5f  5g  5d

X.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
X.3      2  A *A  .  2   A  *A  2   A  *A  .  .  2   A  *A   A  *A
X.4      2 *A  A  .  2  *A   A  2  *A   A  .  .  2  *A   A  *A   A
X.5      4  4  4 -2  .   .   .  1   1   1  1  . -1  -1  -1  -1  -1
X.6      4  4  4  2  .   .   .  1   1   1 -1  . -1  -1  -1  -1  -1
X.7      5  5  5 -1  1   1   1 -1  -1  -1 -1  1  .   .   .   .   .
X.8      5  5  5  1  1   1   1 -1  -1  -1  1 -1  .   .   .   .   .
X.9      6  6  6  . -2  -2  -2  .   .   .  .  .  1   1   1   1   1
X.10     6  B *B  . -2 -*A  -A  .   .   .  .  .  1   E  /F  /E   F
X.11     6  B *B  . -2 -*A  -A  .   .   .  .  .  1  /E   F   E  /F
X.12     6 *B  B  . -2  -A -*A  .   .   .  .  .  1   F   E  /F  /E
X.13     6 *B  B  . -2  -A -*A  .   .   .  .  .  1  /F  /E   F   E
X.14     8  C *C  .  .   .   .  2   A  *A  .  . -2  -A -*A  -A -*A
X.15     8 *C  C  .  .   .   .  2  *A   A  .  . -2 -*A  -A -*A  -A
X.16    10  D *D  .  2   A  *A -2  -A -*A  .  .  .   .   .   .   .
X.17    10 *D  D  .  2  *A   A -2 -*A  -A  .  .  .   .   .   .   .

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