Group action invariants
Degree $n$: | $37$ | |
Transitive number $t$: | $6$ | |
Group: | $C_{37}:C_{9}$ | |
Parity: | $1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (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), (1,16,34,26,9,33,10,12,7)(2,32,31,15,18,29,20,24,14)(3,11,28,4,27,25,30,36,21)(5,6,22,19,8,17,13,23,35) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $9$: $C_9$ 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 $ | $1$ | $1$ | $()$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2, 8,13,11,34,10,27,35,17)( 3,15,25,21,30,19,16,32,33)( 4,22,37,31,26,28, 5, 29,12)( 6,36,24,14,18, 9,20,23, 7)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,10, 8,27,13,35,11,17,34)( 3,19,15,16,25,32,21,33,30)( 4,28,22, 5,37,29,31, 12,26)( 6, 9,36,20,24,23,14, 7,18)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $37$ | $3$ | $( 2,11,27)( 3,21,16)( 4,31, 5)( 6,14,20)( 7,24, 9)( 8,34,35)(10,17,13) (12,37,28)(15,30,32)(18,23,36)(19,33,25)(22,26,29)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,13,34,27,17, 8,11,10,35)( 3,25,30,16,33,15,21,19,32)( 4,37,26, 5,12,22,31, 28,29)( 6,24,18,20, 7,36,14, 9,23)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,17,35,27,10,34,11,13, 8)( 3,33,32,16,19,30,21,25,15)( 4,12,29, 5,28,26,31, 37,22)( 6, 7,23,20, 9,18,14,24,36)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $37$ | $3$ | $( 2,27,11)( 3,16,21)( 4, 5,31)( 6,20,14)( 7, 9,24)( 8,35,34)(10,13,17) (12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,34,17,11,35,13,27, 8,10)( 3,30,33,21,32,25,16,15,19)( 4,26,12,31,29,37, 5, 22,28)( 6,18, 7,14,23,24,20,36, 9)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,35,10,11, 8,17,27,34,13)( 3,32,19,21,15,33,16,30,25)( 4,29,28,31,22,12, 5, 26,37)( 6,23, 9,14,36, 7,20,18,24)$ |
$ 37 $ | $9$ | $37$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$ |
$ 37 $ | $9$ | $37$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$ |
$ 37 $ | $9$ | $37$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$ |
$ 37 $ | $9$ | $37$ | $( 1, 6,11,16,21,26,31,36, 4, 9,14,19,24,29,34, 2, 7,12,17,22,27,32,37, 5,10, 15,20,25,30,35, 3, 8,13,18,23,28,33)$ |
Group invariants
Order: | $333=3^{2} \cdot 37$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [333, 3] |
Character table: |
3 2 2 2 2 2 2 2 2 2 . . . . 37 1 . . . . . . . . 1 1 1 1 1a 9a 9b 3a 9c 9d 3b 9e 9f 37a 37b 37c 37d 2P 1a 9c 9a 3b 9e 9f 3a 9d 9b 37b 37c 37d 37a 3P 1a 3a 3b 1a 3b 3b 1a 3a 3a 37c 37d 37a 37b 5P 1a 9b 9f 3b 9a 9e 3a 9c 9d 37d 37a 37b 37c 7P 1a 9f 9d 3a 9b 9c 3b 9a 9e 37a 37b 37c 37d 11P 1a 9c 9a 3b 9e 9f 3a 9d 9b 37c 37d 37a 37b 13P 1a 9e 9c 3a 9d 9b 3b 9f 9a 37d 37a 37b 37c 17P 1a 9d 9e 3b 9f 9a 3a 9b 9c 37d 37a 37b 37c 19P 1a 9a 9b 3a 9c 9d 3b 9e 9f 37d 37a 37b 37c 23P 1a 9b 9f 3b 9a 9e 3a 9c 9d 37d 37a 37b 37c 29P 1a 9c 9a 3b 9e 9f 3a 9d 9b 37b 37c 37d 37a 31P 1a 9e 9c 3a 9d 9b 3b 9f 9a 37b 37c 37d 37a 37P 1a 9a 9b 3a 9c 9d 3b 9e 9f 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 A /A 1 /A /A 1 A A 1 1 1 1 X.3 1 /A A 1 A A 1 /A /A 1 1 1 1 X.4 1 B /D A C /B /A D /C 1 1 1 1 X.5 1 /B D /A /C B A /D C 1 1 1 1 X.6 1 C B /A D /C A /B /D 1 1 1 1 X.7 1 D C A /B /D /A /C B 1 1 1 1 X.8 1 /D /C /A B D A C /B 1 1 1 1 X.9 1 /C /B A /D C /A B D 1 1 1 1 X.10 9 . . . . . . . . E F /E /F X.11 9 . . . . . . . . F /E /F E X.12 9 . . . . . . . . /F E F /E X.13 9 . . . . . . . . /E /F E F A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = -E(9)^2-E(9)^5 C = E(9)^7 D = E(9)^5 E = E(37)+E(37)^7+E(37)^9+E(37)^10+E(37)^12+E(37)^16+E(37)^26+E(37)^33+E(37)^34 F = E(37)^2+E(37)^14+E(37)^15+E(37)^18+E(37)^20+E(37)^24+E(37)^29+E(37)^31+E(37)^32 |