Group action invariants
| Degree $n$ : | $37$ | |
| Transitive number $t$ : | $5$ | |
| Group : | $C_{37}:C_{6}$ | |
| Parity: | $1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| 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), (1,27,26,36,10,11)(2,17,15,35,20,22)(3,7,4,34,30,33)(5,24,19,32,13,18)(6,14,8,31,23,29)(9,21,12,28,16,25) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $37$ | $3$ | $( 2,11,27)( 3,21,16)( 4,31, 5)( 6,14,20)( 7,24, 9)( 8,34,35)(10,17,13) (12,37,28)(15,30,32)(18,23,36)(19,33,25)(22,26,29)$ |
| $ 6, 6, 6, 6, 6, 6, 1 $ | $37$ | $6$ | $( 2,12,11,37,27,28)( 3,23,21,36,16,18)( 4,34,31,35, 5, 8)( 6,19,14,33,20,25) ( 7,30,24,32, 9,15)(10,26,17,29,13,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $37$ | $3$ | $( 2,27,11)( 3,16,21)( 4, 5,31)( 6,20,14)( 7, 9,24)( 8,35,34)(10,13,17) (12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)$ |
| $ 6, 6, 6, 6, 6, 6, 1 $ | $37$ | $6$ | $( 2,28,27,37,11,12)( 3,18,16,36,21,23)( 4, 8, 5,35,31,34)( 6,25,20,33,14,19) ( 7,15, 9,32,24,30)(10,22,13,29,17,26)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $37$ | $2$ | $( 2,37)( 3,36)( 4,35)( 5,34)( 6,33)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,27) (13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
| $ 37 $ | $6$ | $37$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$ |
| $ 37 $ | $6$ | $37$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$ |
| $ 37 $ | $6$ | $37$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$ |
| $ 37 $ | $6$ | $37$ | $( 1, 6,11,16,21,26,31,36, 4, 9,14,19,24,29,34, 2, 7,12,17,22,27,32,37, 5,10, 15,20,25,30,35, 3, 8,13,18,23,28,33)$ |
| $ 37 $ | $6$ | $37$ | $( 1, 7,13,19,25,31,37, 6,12,18,24,30,36, 5,11,17,23,29,35, 4,10,16,22,28,34, 3, 9,15,21,27,33, 2, 8,14,20,26,32)$ |
| $ 37 $ | $6$ | $37$ | $( 1,10,19,28,37, 9,18,27,36, 8,17,26,35, 7,16,25,34, 6,15,24,33, 5,14,23,32, 4,13,22,31, 3,12,21,30, 2,11,20,29)$ |
Group invariants
| Order: | $222=2 \cdot 3 \cdot 37$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [222, 1] |
| Character table: |
2 1 1 1 1 1 1 . . . . . .
3 1 1 1 1 1 1 . . . . . .
37 1 . . . . . 1 1 1 1 1 1
1a 3a 6a 3b 6b 2a 37a 37b 37c 37d 37e 37f
2P 1a 3b 3a 3a 3b 1a 37b 37c 37e 37a 37f 37d
3P 1a 1a 2a 1a 2a 2a 37c 37e 37f 37b 37d 37a
5P 1a 3b 6b 3a 6a 2a 37d 37a 37b 37f 37c 37e
7P 1a 3a 6a 3b 6b 2a 37c 37e 37f 37b 37d 37a
11P 1a 3b 6b 3a 6a 2a 37a 37b 37c 37d 37e 37f
13P 1a 3a 6a 3b 6b 2a 37d 37a 37b 37f 37c 37e
17P 1a 3b 6b 3a 6a 2a 37b 37c 37e 37a 37f 37d
19P 1a 3a 6a 3b 6b 2a 37d 37a 37b 37f 37c 37e
23P 1a 3b 6b 3a 6a 2a 37e 37f 37d 37c 37a 37b
29P 1a 3b 6b 3a 6a 2a 37e 37f 37d 37c 37a 37b
31P 1a 3a 6a 3b 6b 2a 37e 37f 37d 37c 37a 37b
37P 1a 3a 6a 3b 6b 2a 1a 1a 1a 1a 1a 1a
X.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
X.3 1 A -/A /A -A -1 1 1 1 1 1 1
X.4 1 /A -A A -/A -1 1 1 1 1 1 1
X.5 1 A /A /A A 1 1 1 1 1 1 1
X.6 1 /A A A /A 1 1 1 1 1 1 1
X.7 6 . . . . . B G F C E D
X.8 6 . . . . . C B G D F E
X.9 6 . . . . . D C B E G F
X.10 6 . . . . . E D C F B G
X.11 6 . . . . . F E D G C B
X.12 6 . . . . . G F E B D C
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = E(37)^3+E(37)^4+E(37)^7+E(37)^30+E(37)^33+E(37)^34
C = E(37)^2+E(37)^15+E(37)^17+E(37)^20+E(37)^22+E(37)^35
D = E(37)+E(37)^10+E(37)^11+E(37)^26+E(37)^27+E(37)^36
E = E(37)^5+E(37)^13+E(37)^18+E(37)^19+E(37)^24+E(37)^32
F = E(37)^9+E(37)^12+E(37)^16+E(37)^21+E(37)^25+E(37)^28
G = E(37)^6+E(37)^8+E(37)^14+E(37)^23+E(37)^29+E(37)^31
|