Properties

Label 25T46
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 $t$ :  $46$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,4,5)(6,7,8,9,10)(11,22,18,14,25,16,12,23,19,15,21,17,13,24,20), (1,16,6,11,21)(2,20,7,15,22,5,17,10,12,25)(3,19,8,14,23,4,18,9,13,24)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
10:  $D_{5}$
60:  $A_5$
120:  $A_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$, $A_5$

Low degree siblings

30T128

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, 1, 1, 1, 1, 1 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 10, 10, 5 $ $30$ $10$ $( 1, 7, 3, 9, 5, 6, 2, 8, 4,10)(11,17,13,19,15,16,12,18,14,20)(21,22,23,24,25)$
$ 10, 10, 5 $ $30$ $10$ $( 1, 8, 5, 7, 4, 6, 3,10, 2, 9)(11,18,15,17,14,16,13,20,12,19)(21,23,25,22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $75$ $2$ $( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(11,16)(12,20)(13,19)(14,18)(15,17)(22,25) (23,24)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 15, 5, 5 $ $40$ $15$ $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,17,18,19,20)(21,22,23,24,25)$
$ 15, 5, 5 $ $40$ $15$ $( 1, 8,15, 2, 9,11, 3,10,12, 4, 6,13, 5, 7,14)(16,18,20,17,19)(21,23,25,22,24)$
$ 6, 6, 3, 2, 2, 2, 2, 1, 1 $ $100$ $6$ $( 1, 6,11)( 2,10,12, 5, 7,15)( 3, 9,13, 4, 8,14)(17,20)(18,19)(22,25)(23,24)$
$ 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$
$ 10, 10, 5 $ $60$ $10$ $( 1, 6,11,16,21)( 2,10,12,20,22, 5, 7,15,17,25)( 3, 9,13,19,23, 4, 8,14,18,24)$
$ 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 6,11,21,16)( 2, 7,12,22,17)( 3, 8,13,23,18)( 4, 9,14,24,19) ( 5,10,15,25,20)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 7,13,24,20)( 2, 8,14,25,16)( 3, 9,15,21,17)( 4,10,11,22,18) ( 5, 6,12,23,19)$
$ 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 8,15,22,19)( 2, 9,11,23,20)( 3,10,12,24,16)( 4, 6,13,25,17) ( 5, 7,14,21,18)$
$ 10, 10, 5 $ $60$ $10$ $( 1, 6,11,21,16)( 2,10,12,25,17, 5, 7,15,22,20)( 3, 9,13,24,18, 4, 8,14,23,19)$

Group invariants

Order:  $600=2^{3} \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [600, 146]
Character table:   
      2  3  2  2  3  3   2   2  3  1   .   .  1   1   .   .   1   1   .   .
      3  1  1  1  1  .   .   .  .  1   1   1  1   .   .   .   .   .   .   .
      5  2  2  2  1  1   1   1  .  1   1   1  .   2   2   2   1   2   2   2

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

X.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
X.3      2  A *A  .  2   A  *A  .  2   A  *A  .   2   A  *A   .   2   A  *A
X.4      2 *A  A  .  2  *A   A  .  2  *A   A  .   2  *A   A   .   2  *A   A
X.5      3  3  3 -3 -1  -1  -1  1  .   .   .  .  -A  -A  -A   A -*A -*A -*A
X.6      3  3  3 -3 -1  -1  -1  1  .   .   .  . -*A -*A -*A  *A  -A  -A  -A
X.7      3  3  3  3 -1  -1  -1 -1  .   .   .  .  -A  -A  -A  -A -*A -*A -*A
X.8      3  3  3  3 -1  -1  -1 -1  .   .   .  . -*A -*A -*A -*A  -A  -A  -A
X.9      4  4  4 -4  .   .   .  .  1   1   1 -1  -1  -1  -1   1  -1  -1  -1
X.10     4  4  4  4  .   .   .  .  1   1   1  1  -1  -1  -1  -1  -1  -1  -1
X.11     5  5  5 -5  1   1   1 -1 -1  -1  -1  1   .   .   .   .   .   .   .
X.12     5  5  5  5  1   1   1  1 -1  -1  -1 -1   .   .   .   .   .   .   .
X.13     6  B *B  . -2 -*A  -A  .  .   .   .  .   E   1  *F   .  *E   F   1
X.14     6 *B  B  . -2  -A -*A  .  .   .   .  .  *E   1   F   .   E  *F   1
X.15     6  B *B  . -2 -*A  -A  .  .   .   .  .  *E   F   1   .   E   1  *F
X.16     6 *B  B  . -2  -A -*A  .  .   .   .  .   E  *F   1   .  *E   1   F
X.17     8  C *C  .  .   .   .  .  2   A  *A  .  -2  -A -*A   .  -2  -A -*A
X.18     8 *C  C  .  .   .   .  .  2  *A   A  .  -2 -*A  -A   .  -2 -*A  -A
X.19    10  D *D  .  2   A  *A  . -2  -A -*A  .   .   .   .   .   .   .   .
X.20    10 *D  D  .  2  *A   A  . -2 -*A  -A  .   .   .   .   .   .   .   .

      2   1
      3   .
      5   1

        10d
     2P  5c
     3P 10c
     5P  2a
     7P 10c
    11P 10d
    13P 10c

X.1       1
X.2      -1
X.3       .
X.4       .
X.5      *A
X.6       A
X.7     -*A
X.8      -A
X.9       1
X.10     -1
X.11      .
X.12      .
X.13      .
X.14      .
X.15      .
X.16      .
X.17      .
X.18      .
X.19      .
X.20      .

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 = -2*E(5)-2*E(5)^4
  = 1-Sqrt(5) = 1-r5
F = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4
  = (-3-Sqrt(5))/2 = -2-b5