Group action invariants
| Degree $n$: | $37$ | |
| Transitive number $t$: | $7$ | |
| Group: | $C_{37}:C_{12}$ | |
| 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), (1,8,27,31,26,23,36,29,10,6,11,14)(2,16,17,25,15,9,35,21,20,12,22,28)(3,24,7,19,4,32,34,13,30,18,33,5) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $C_6$ $12$: $C_{12}$ 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 $ | $1$ | $1$ | $()$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $37$ | $4$ | $( 2, 7,37,32)( 3,13,36,26)( 4,19,35,20)( 5,25,34,14)( 6,31,33, 8)( 9,12,30,27) (10,18,29,21)(11,24,28,15)(16,17,23,22)$ |
| $ 12, 12, 12, 1 $ | $37$ | $12$ | $( 2, 9,28,32,27,24,37,30,11, 7,12,15)( 3,17,18,26,16,10,36,22,21,13,23,29) ( 4,25, 8,20, 5,33,35,14,31,19,34, 6)$ |
| $ 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 $ | $37$ | $6$ | $( 2,12,11,37,27,28)( 3,23,21,36,16,18)( 4,34,31,35, 5, 8)( 6,19,14,33,20,25) ( 7,30,24,32, 9,15)(10,26,17,29,13,22)$ |
| $ 12, 12, 12, 1 $ | $37$ | $12$ | $( 2,15,12, 7,11,30,37,24,27,32,28, 9)( 3,29,23,13,21,22,36,10,16,26,18,17) ( 4, 6,34,19,31,14,35,33, 5,20, 8,25)$ |
| $ 12, 12, 12, 1 $ | $37$ | $12$ | $( 2,24,12,32,11, 9,37,15,27, 7,28,30)( 3,10,23,26,21,17,36,29,16,13,18,22) ( 4,33,34,20,31,25,35, 6, 5,19, 8,14)$ |
| $ 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 $ | $37$ | $6$ | $( 2,28,27,37,11,12)( 3,18,16,36,21,23)( 4, 8, 5,35,31,34)( 6,25,20,33,14,19) ( 7,15, 9,32,24,30)(10,22,13,29,17,26)$ |
| $ 12, 12, 12, 1 $ | $37$ | $12$ | $( 2,30,28, 7,27,15,37, 9,11,32,12,24)( 3,22,18,13,16,29,36,17,21,26,23,10) ( 4,14, 8,19, 5, 6,35,25,31,20,34,33)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ | $37$ | $4$ | $( 2,32,37, 7)( 3,26,36,13)( 4,20,35,19)( 5,14,34,25)( 6, 8,33,31)( 9,27,30,12) (10,21,29,18)(11,15,28,24)(16,22,23,17)$ |
| $ 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 $ | $12$ | $37$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$ |
| $ 37 $ | $12$ | $37$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$ |
| $ 37 $ | $12$ | $37$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$ |
Group invariants
| Order: | $444=2^{2} \cdot 3 \cdot 37$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [444, 7] |
| Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 . . .
3 1 1 1 1 1 1 1 1 1 1 1 1 . . .
37 1 . . . . . . . . . . . 1 1 1
1a 4a 12a 3a 6a 12b 12c 3b 6b 12d 4b 2a 37a 37b 37c
2P 1a 2a 6b 3b 3a 6a 6a 3a 3b 6b 2a 1a 37b 37c 37a
3P 1a 4b 4b 1a 2a 4a 4b 1a 2a 4a 4a 2a 37c 37a 37b
5P 1a 4a 12c 3b 6b 12d 12a 3a 6a 12b 4b 2a 37c 37a 37b
7P 1a 4b 12d 3a 6a 12c 12b 3b 6b 12a 4a 2a 37c 37a 37b
11P 1a 4b 12b 3b 6b 12a 12d 3a 6a 12c 4a 2a 37a 37b 37c
13P 1a 4a 12a 3a 6a 12b 12c 3b 6b 12d 4b 2a 37c 37a 37b
17P 1a 4a 12c 3b 6b 12d 12a 3a 6a 12b 4b 2a 37b 37c 37a
19P 1a 4b 12d 3a 6a 12c 12b 3b 6b 12a 4a 2a 37c 37a 37b
23P 1a 4b 12b 3b 6b 12a 12d 3a 6a 12c 4a 2a 37a 37b 37c
29P 1a 4a 12c 3b 6b 12d 12a 3a 6a 12b 4b 2a 37a 37b 37c
31P 1a 4b 12d 3a 6a 12c 12b 3b 6b 12a 4a 2a 37a 37b 37c
37P 1a 4a 12a 3a 6a 12b 12c 3b 6b 12d 4b 2a 1a 1a 1a
X.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
X.3 1 -1 B -/B -B /B /B -B -/B B -1 1 1 1 1
X.4 1 -1 /B -B -/B B B -/B -B /B -1 1 1 1 1
X.5 1 1 -/B -B -/B -B -B -/B -B -/B 1 1 1 1 1
X.6 1 1 -B -/B -B -/B -/B -B -/B -B 1 1 1 1 1
X.7 1 A A 1 -1 -A A 1 -1 -A -A -1 1 1 1
X.8 1 -A -A 1 -1 A -A 1 -1 A A -1 1 1 1
X.9 1 A C -/B B /C -/C -B /B -C -A -1 1 1 1
X.10 1 A -/C -B /B -C C -/B B /C -A -1 1 1 1
X.11 1 -A /C -B /B C -C -/B B -/C A -1 1 1 1
X.12 1 -A -C -/B B -/C /C -B /B C A -1 1 1 1
X.13 12 . . . . . . . . . . . D F E
X.14 12 . . . . . . . . . . . E D F
X.15 12 . . . . . . . . . . . F E D
A = -E(4)
= -Sqrt(-1) = -i
B = -E(3)
= (1-Sqrt(-3))/2 = -b3
C = -E(12)^7
D = E(37)+E(37)^6+E(37)^8+E(37)^10+E(37)^11+E(37)^14+E(37)^23+E(37)^26+E(37)^27+E(37)^29+E(37)^31+E(37)^36
E = E(37)^3+E(37)^4+E(37)^5+E(37)^7+E(37)^13+E(37)^18+E(37)^19+E(37)^24+E(37)^30+E(37)^32+E(37)^33+E(37)^34
F = E(37)^2+E(37)^9+E(37)^12+E(37)^15+E(37)^16+E(37)^17+E(37)^20+E(37)^21+E(37)^22+E(37)^25+E(37)^28+E(37)^35
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