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
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