Group action invariants
Degree $n$: | $25$ | |
Transitive number $t$: | $50$ | |
Parity: | $-1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (2,23,4,12,5,9,3,20)(6,17,16,24,21,15,11,8)(7,14,19,10,25,18,13,22), (2,15,6,23)(3,24,11,20)(4,8,16,12)(5,17,21,9)(10,14,22,18), (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) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ $32$: $C_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Low degree siblings
10T33, 20T155, 20T161, 20T167, 20T169, 40T874, 40T875, 40T876, 40T877, 40T878, 40T879, 40T880, 40T881, 40T882, 40T883Siblings 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 $ | $16$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
$ 5, 5, 5, 5, 5 $ | $8$ | $5$ | $( 1,14,22,10,18)( 2,15,23, 6,19)( 3,11,24, 7,20)( 4,12,25, 8,16) ( 5,13,21, 9,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$ |
$ 8, 8, 8, 1 $ | $100$ | $8$ | $( 2,23, 4,12, 5, 9, 3,20)( 6,17,16,24,21,15,11, 8)( 7,14,19,10,25,18,13,22)$ |
$ 8, 8, 8, 1 $ | $100$ | $8$ | $( 2,12, 3,23, 5,20, 4, 9)( 6,24,11,17,21, 8,16,15)( 7,10,13,14,25,22,19,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 2, 6)( 3,11)( 4,16)( 5,21)( 8,12)( 9,17)(10,22)(14,18)(15,23)(20,24)$ |
$ 10, 10, 5 $ | $40$ | $10$ | $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $50$ | $4$ | $( 2,16, 5,11)( 3, 6, 4,21)( 7,19,25,13)( 8, 9,24,23)(10,14,22,18)(12,17,20,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $20$ | $2$ | $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$ |
$ 10, 10, 5 $ | $80$ | $10$ | $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $100$ | $4$ | $( 2, 4, 5, 3)( 6,11,21,16)( 7,14,25,18)( 8,12,24,20)( 9,15,23,17)(10,13,22,19)$ |
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ | $10$ | $4$ | $( 2,23, 6,15)( 3,20,11,24)( 4,12,16, 8)( 5, 9,21,17)(10,18,22,14)$ |
$ 20, 5 $ | $40$ | $20$ | $( 1, 2,24, 4,13,14, 6,11,25,21,18,23, 7, 8, 5,10,19,20,12,17)( 3,16, 9,22,15)$ |
$ 4, 4, 4, 4, 4, 2, 2, 1 $ | $50$ | $4$ | $( 2, 9, 6,17)( 3,12,11, 8)( 4,20,16,24)( 5,23,21,15)( 7,25)(10,14,22,18) (13,19)$ |
$ 4, 4, 4, 4, 4, 2, 2, 1 $ | $50$ | $4$ | $( 2,12,21,24)( 3,23,16,17)( 4, 9,11,15)( 5,20, 6, 8)( 7,19,25,13)(10,22) (14,18)$ |
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ | $10$ | $4$ | $( 2,20,21, 8)( 3, 9,16,15)( 4,23,11,17)( 5,12, 6,24)( 7,13,25,19)$ |
$ 20, 5 $ | $40$ | $20$ | $( 1, 2,16,11,18,19, 8, 3,10, 6,25,20,22,23,12, 7,14,15, 4,24)( 5,13,21, 9,17)$ |
Group invariants
Order: | $800=2^{5} \cdot 5^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [800, 1191] |
Character table: |
2 5 1 2 5 5 5 3 3 4 2 4 3 1 3 4 2 4 4 4 2 5 2 2 2 . . . . . 1 1 . 1 1 . 1 1 . . 1 1 1a 5a 5b 2a 4a 4b 8a 8b 2b 10a 4c 2c 10b 4d 4e 20a 4f 4g 4h 20b 2P 1a 5a 5b 1a 2a 2a 4a 4b 1a 5b 2a 1a 5a 2a 2b 10a 2b 2b 2b 10a 3P 1a 5a 5b 2a 4b 4a 8b 8a 2b 10a 4c 2c 10b 4d 4h 20b 4g 4f 4e 20a 5P 1a 1a 1a 2a 4a 4b 8a 8b 2b 2b 4c 2c 2c 4d 4e 4e 4f 4g 4h 4h 7P 1a 5a 5b 2a 4b 4a 8b 8a 2b 10a 4c 2c 10b 4d 4h 20b 4g 4f 4e 20a 11P 1a 5a 5b 2a 4b 4a 8b 8a 2b 10a 4c 2c 10b 4d 4h 20b 4g 4f 4e 20a 13P 1a 5a 5b 2a 4a 4b 8a 8b 2b 10a 4c 2c 10b 4d 4e 20a 4f 4g 4h 20b 17P 1a 5a 5b 2a 4a 4b 8a 8b 2b 10a 4c 2c 10b 4d 4e 20a 4f 4g 4h 20b 19P 1a 5a 5b 2a 4b 4a 8b 8a 2b 10a 4c 2c 10b 4d 4h 20b 4g 4f 4e 20a X.1 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 1 X.3 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 X.4 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 X.5 1 1 1 1 -1 -1 B -B -1 -1 1 -1 -1 1 -B -B -B B B B X.6 1 1 1 1 -1 -1 -B B -1 -1 1 -1 -1 1 B B B -B -B -B X.7 1 1 1 1 -1 -1 B -B -1 -1 1 1 1 -1 B B B -B -B -B X.8 1 1 1 1 -1 -1 -B B -1 -1 1 1 1 -1 -B -B -B B B B X.9 2 2 2 2 2 2 . . -2 -2 -2 . . . . . . . . . X.10 2 2 2 2 -2 -2 . . 2 2 -2 . . . . . . . . . X.11 2 2 2 -2 A -A . . . . . . . . C C -C -/C /C /C X.12 2 2 2 -2 -A A . . . . . . . . /C /C -/C -C C C X.13 2 2 2 -2 -A A . . . . . . . . -/C -/C /C C -C -C X.14 2 2 2 -2 A -A . . . . . . . . -C -C C /C -/C -/C X.15 8 -2 3 . . . . . 4 -1 . . . . -4 1 . . -4 1 X.16 8 -2 3 . . . . . 4 -1 . . . . 4 -1 . . 4 -1 X.17 8 -2 3 . . . . . -4 1 . . . . D -B . . -D B X.18 8 -2 3 . . . . . -4 1 . . . . -D B . . D -B X.19 16 1 -4 . . . . . . . . -4 1 . . . . . . . X.20 16 1 -4 . . . . . . . . 4 -1 . . . . . . . A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(4) = -Sqrt(-1) = -i C = 1-E(4) = 1-Sqrt(-1) = 1-i D = -4*E(4) = -4*Sqrt(-1) = -4i |