Group action invariants
Degree $n$: | $25$ | |
Transitive number $t$: | $44$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
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) |
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
Low degree siblings
30T144Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 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 |