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 |