Group action invariants
Degree $n$: | $41$ | |
Transitive number $t$: | $4$ | |
Group: | $C_{41}:C_{5}$ | |
Parity: | $1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,10,18,16,37)(2,20,36,32,33)(3,30,13,7,29)(4,40,31,23,25)(5,9,8,39,21)(6,19,26,14,17)(11,28,34,12,38)(15,27,24,35,22), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $5$: $C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,11,19,17,38)( 3,21,37,33,34)( 4,31,14, 8,30)( 5,41,32,24,26) ( 6,10, 9,40,22)( 7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,17,11,38,19)( 3,33,21,34,37)( 4, 8,31,30,14)( 5,24,41,26,32) ( 6,40,10,22, 9)( 7,15,20,18,27)(12,13,29,39,35)(16,36,28,23,25)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,19,38,11,17)( 3,37,34,21,33)( 4,14,30,31, 8)( 5,32,26,41,24) ( 6, 9,22,10,40)( 7,27,18,20,15)(12,35,39,29,13)(16,25,23,28,36)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ | $41$ | $5$ | $( 2,38,17,19,11)( 3,34,33,37,21)( 4,30, 8,14,31)( 5,26,24,32,41) ( 6,22,40, 9,10)( 7,18,15,27,20)(12,39,13,35,29)(16,23,36,25,28)$ |
$ 41 $ | $5$ | $41$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$ |
$ 41 $ | $5$ | $41$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8, 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$ |
$ 41 $ | $5$ | $41$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40, 2, 5, 8,11,14,17,20,23,26,29,32, 35,38,41, 3, 6, 9,12,15,18,21,24,27,30,33,36,39)$ |
$ 41 $ | $5$ | $41$ | $( 1, 5, 9,13,17,21,25,29,33,37,41, 4, 8,12,16,20,24,28,32,36,40, 3, 7,11,15, 19,23,27,31,35,39, 2, 6,10,14,18,22,26,30,34,38)$ |
$ 41 $ | $5$ | $41$ | $( 1, 6,11,16,21,26,31,36,41, 5,10,15,20,25,30,35,40, 4, 9,14,19,24,29,34,39, 3, 8,13,18,23,28,33,38, 2, 7,12,17,22,27,32,37)$ |
$ 41 $ | $5$ | $41$ | $( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22, 28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$ |
$ 41 $ | $5$ | $41$ | $( 1,12,23,34, 4,15,26,37, 7,18,29,40,10,21,32, 2,13,24,35, 5,16,27,38, 8,19, 30,41,11,22,33, 3,14,25,36, 6,17,28,39, 9,20,31)$ |
$ 41 $ | $5$ | $41$ | $( 1,16,31, 5,20,35, 9,24,39,13,28, 2,17,32, 6,21,36,10,25,40,14,29, 3,18,33, 7,22,37,11,26,41,15,30, 4,19,34, 8,23,38,12,27)$ |
Group invariants
Order: | $205=5 \cdot 41$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [205, 1] |
Character table: |
5 1 1 1 1 1 . . . . . . . . 41 1 . . . . 1 1 1 1 1 1 1 1 1a 5a 5b 5c 5d 41a 41b 41c 41d 41e 41f 41g 41h 2P 1a 5c 5a 5d 5b 41b 41d 41f 41e 41a 41g 41h 41c 3P 1a 5b 5d 5a 5c 41c 41f 41e 41g 41h 41a 41b 41d 5P 1a 1a 1a 1a 1a 41e 41a 41h 41b 41d 41c 41f 41g 7P 1a 5c 5a 5d 5b 41c 41f 41e 41g 41h 41a 41b 41d 11P 1a 5a 5b 5c 5d 41g 41h 41b 41c 41f 41d 41e 41a 13P 1a 5b 5d 5a 5c 41c 41f 41e 41g 41h 41a 41b 41d 17P 1a 5c 5a 5d 5b 41f 41g 41a 41h 41c 41b 41d 41e 19P 1a 5d 5c 5b 5a 41f 41g 41a 41h 41c 41b 41d 41e 23P 1a 5b 5d 5a 5c 41d 41e 41g 41a 41b 41h 41c 41f 29P 1a 5d 5c 5b 5a 41c 41f 41e 41g 41h 41a 41b 41d 31P 1a 5a 5b 5c 5d 41d 41e 41g 41a 41b 41h 41c 41f 37P 1a 5c 5a 5d 5b 41a 41b 41c 41d 41e 41f 41g 41h 41P 1a 5a 5b 5c 5d 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A /B B /A 1 1 1 1 1 1 1 1 X.3 1 B A /A /B 1 1 1 1 1 1 1 1 X.4 1 /B /A A B 1 1 1 1 1 1 1 1 X.5 1 /A B /B A 1 1 1 1 1 1 1 1 X.6 5 . . . . C /F /E /C F D E /D X.7 5 . . . . /C F E C /F /D /E D X.8 5 . . . . D E C /D /E /F /C F X.9 5 . . . . E /D /F /E D /C F C X.10 5 . . . . F C /D /F /C /E D E X.11 5 . . . . /E D F E /D C /F /C X.12 5 . . . . /D /E /C D E F C /F X.13 5 . . . . /F /C D F C E /D /E A = E(5)^4 B = E(5)^3 C = E(41)^3+E(41)^7+E(41)^13+E(41)^29+E(41)^30 D = E(41)+E(41)^10+E(41)^16+E(41)^18+E(41)^37 E = E(41)^2+E(41)^20+E(41)^32+E(41)^33+E(41)^36 F = E(41)^15+E(41)^22+E(41)^24+E(41)^27+E(41)^35 |