Group action invariants
Degree $n$: | $25$ | |
Transitive number $t$: | $33$ | |
Group: | $C_5^2:F_5$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,8,12,17,25)(2,9,13,18,21)(3,10,14,19,22)(4,6,15,20,23)(5,7,11,16,24), (1,18,21,6)(2,17,22,10)(3,16,23,9)(4,20,24,8)(5,19,25,7)(11,13)(14,15), (1,8,5,7,4,6,3,10,2,9)(11,22,15,21,14,25,13,24,12,23)(16,18,20,17,19) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ x 6 $100$: 25T9 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Low degree siblings
25T33 x 5Siblings 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, 1, 1, 1, 1, 1 $ | $20$ | $5$ | $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $25$ | $2$ | $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,17)(12,18)(13,19)(14,20)(15,16)$ |
$ 4, 4, 4, 4, 4, 2, 2, 1 $ | $125$ | $4$ | $( 2, 5)( 3, 4)( 6,11,25,20)( 7,15,21,19)( 8,14,22,18)( 9,13,23,17) (10,12,24,16)$ |
$ 4, 4, 4, 4, 4, 2, 2, 1 $ | $125$ | $4$ | $( 2, 5)( 3, 4)( 6,16,22,13)( 7,20,23,12)( 8,19,24,11)( 9,18,25,15) (10,17,21,14)$ |
$ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
$ 10, 10, 5 $ | $50$ | $10$ | $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,16,13,18,15,20,12,17,14,19)$ |
$ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ |
$ 10, 10, 5 $ | $50$ | $10$ | $( 1, 3, 5, 2, 4)( 6,21,10,25, 9,24, 8,23, 7,22)(11,20,15,19,14,18,13,17,12,16)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,11,20,25)( 2, 7,12,16,21)( 3, 8,13,17,22)( 4, 9,14,18,23) ( 5,10,15,19,24)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,12,18,21)( 2, 7,13,19,22)( 3, 8,14,20,23)( 4, 9,15,16,24) ( 5,10,11,17,25)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,13,16,22)( 2, 7,14,17,23)( 3, 8,15,18,24)( 4, 9,11,19,25) ( 5,10,12,20,21)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,14,19,23)( 2, 7,15,20,24)( 3, 8,11,16,25)( 4, 9,12,17,21) ( 5,10,13,18,22)$ |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,15,17,24)( 2, 7,11,18,25)( 3, 8,12,19,21)( 4, 9,13,20,22) ( 5,10,14,16,23)$ |
Group invariants
Order: | $500=2^{2} \cdot 5^{3}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [500, 23] |
Character table: |
2 2 . 2 2 2 1 1 1 1 . . . . . 5 3 2 1 . . 3 1 3 1 2 2 2 2 2 1a 5a 2a 4a 4b 5b 10a 5c 10b 5d 5e 5f 5g 5h 2P 1a 5a 1a 2a 2a 5c 5c 5b 5b 5d 5e 5f 5g 5h 3P 1a 5a 2a 4b 4a 5c 10b 5b 10a 5d 5e 5f 5g 5h 5P 1a 1a 2a 4a 4b 1a 2a 1a 2a 1a 1a 1a 1a 1a 7P 1a 5a 2a 4b 4a 5c 10b 5b 10a 5d 5e 5f 5g 5h 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 A -A 1 -1 1 -1 1 1 1 1 1 X.4 1 1 -1 -A A 1 -1 1 -1 1 1 1 1 1 X.5 4 4 . . . 4 . 4 . -1 -1 -1 -1 -1 X.6 4 -1 . . . 4 . 4 . 4 -1 -1 -1 -1 X.7 4 -1 . . . 4 . 4 . -1 4 -1 -1 -1 X.8 4 -1 . . . 4 . 4 . -1 -1 4 -1 -1 X.9 4 -1 . . . 4 . 4 . -1 -1 -1 -1 4 X.10 4 -1 . . . 4 . 4 . -1 -1 -1 4 -1 X.11 10 . -2 . . B C *B *C . . . . . X.12 10 . -2 . . *B *C B C . . . . . X.13 10 . 2 . . B -C *B -*C . . . . . X.14 10 . 2 . . *B -*C B -C . . . . . A = -E(4) = -Sqrt(-1) = -i B = 5*E(5)^2+5*E(5)^3 = (-5-5*Sqrt(5))/2 = -5-5b5 C = -E(5)^2-E(5)^3 = (1+Sqrt(5))/2 = 1+b5 |