Group action invariants
Degree $n$: | $43$ | |
Transitive number $t$: | $6$ | |
Group: | $C_{43}:C_{14}$ | |
Parity: | $-1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $1$ | |
Generators: | (1,27,41,32,4,22,35,42,16,2,11,39,21,8)(3,38,37,10,12,23,19,40,5,6,33,31,20,24)(7,17,29,9,28,25,30,36,26,14,34,15,18,13), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $7$: $C_7$ $14$: $C_{14}$ 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, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 43 $ | $14$ | $43$ | $( 1,35,26,17, 8,42,33,24,15, 6,40,31,22,13, 4,38,29,20,11, 2,36,27,18, 9,43, 34,25,16, 7,41,32,23,14, 5,39,30,21,12, 3,37,28,19,10)$ |
$ 43 $ | $14$ | $43$ | $( 1,17,33, 6,22,38,11,27,43,16,32, 5,21,37,10,26,42,15,31, 4,20,36, 9,25,41, 14,30, 3,19,35, 8,24,40,13,29, 2,18,34, 7,23,39,12,28)$ |
$ 43 $ | $14$ | $43$ | $( 1, 6,11,16,21,26,31,36,41, 3, 8,13,18,23,28,33,38,43, 5,10,15,20,25,30,35, 40, 2, 7,12,17,22,27,32,37,42, 4, 9,14,19,24,29,34,39)$ |
$ 7, 7, 7, 7, 7, 7, 1 $ | $43$ | $7$ | $( 2,42, 5,36,17,12,22)( 3,40, 9,28,33,23,43)( 4,38,13,20, 6,34,21) ( 7,32,25,39,11,24,41)( 8,30,29,31,27,35,19)(10,26,37,15,16,14,18)$ |
$ 7, 7, 7, 7, 7, 7, 1 $ | $43$ | $7$ | $( 2,36,22, 5,12,42,17)( 3,28,43, 9,23,40,33)( 4,20,21,13,34,38, 6) ( 7,39,41,25,24,32,11)( 8,31,19,29,35,30,27)(10,15,18,37,14,26,16)$ |
$ 7, 7, 7, 7, 7, 7, 1 $ | $43$ | $7$ | $( 2, 5,17,22,42,36,12)( 3, 9,33,43,40,28,23)( 4,13, 6,21,38,20,34) ( 7,25,11,41,32,39,24)( 8,29,27,19,30,31,35)(10,37,16,18,26,15,14)$ |
$ 7, 7, 7, 7, 7, 7, 1 $ | $43$ | $7$ | $( 2,22,12,17,36, 5,42)( 3,43,23,33,28, 9,40)( 4,21,34, 6,20,13,38) ( 7,41,24,11,39,25,32)( 8,19,35,27,31,29,30)(10,18,14,16,15,37,26)$ |
$ 7, 7, 7, 7, 7, 7, 1 $ | $43$ | $7$ | $( 2,17,42,12, 5,22,36)( 3,33,40,23, 9,43,28)( 4, 6,38,34,13,21,20) ( 7,11,32,24,25,41,39)( 8,27,30,35,29,19,31)(10,16,26,14,37,18,15)$ |
$ 7, 7, 7, 7, 7, 7, 1 $ | $43$ | $7$ | $( 2,12,36,42,22,17, 5)( 3,23,28,40,43,33, 9)( 4,34,20,38,21, 6,13) ( 7,24,39,32,41,11,25)( 8,35,31,30,19,27,29)(10,14,15,26,18,16,37)$ |
$ 14, 14, 14, 1 $ | $43$ | $14$ | $( 2,28,42,33, 5,23,36,43,17, 3,12,40,22, 9)( 4,39,38,11,13,24,20,41, 6, 7,34, 32,21,25)( 8,18,30,10,29,26,31,37,27,15,35,16,19,14)$ |
$ 14, 14, 14, 1 $ | $43$ | $14$ | $( 2,33,36, 3,22,28, 5,43,12, 9,42,23,17,40)( 4,11,20, 7,21,39,13,41,34,25,38, 24, 6,32)( 8,10,31,15,19,18,29,37,35,14,30,26,27,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $43$ | $2$ | $( 2,43)( 3,42)( 4,41)( 5,40)( 6,39)( 7,38)( 8,37)( 9,36)(10,35)(11,34)(12,33) (13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)$ |
$ 14, 14, 14, 1 $ | $43$ | $14$ | $( 2,23,12,28,36,40,42,43,22,33,17, 9, 5, 3)( 4,24,34,39,20,32,38,41,21,11, 6, 25,13, 7)( 8,26,35,18,31,16,30,37,19,10,27,14,29,15)$ |
$ 14, 14, 14, 1 $ | $43$ | $14$ | $( 2, 9,22,40,12, 3,17,43,36,23, 5,33,42,28)( 4,25,21,32,34, 7, 6,41,20,24,13, 11,38,39)( 8,14,19,16,35,15,27,37,31,26,29,10,30,18)$ |
$ 14, 14, 14, 1 $ | $43$ | $14$ | $( 2, 3, 5, 9,17,33,22,43,42,40,36,28,12,23)( 4, 7,13,25, 6,11,21,41,38,32,20, 39,34,24)( 8,15,29,14,27,10,19,37,30,16,31,18,35,26)$ |
$ 14, 14, 14, 1 $ | $43$ | $14$ | $( 2,40,17,23,42, 9,12,43, 5,28,22, 3,36,33)( 4,32, 6,24,38,25,34,41,13,39,21, 7,20,11)( 8,16,27,26,30,14,35,37,29,18,19,15,31,10)$ |
Group invariants
Order: | $602=2 \cdot 7 \cdot 43$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [602, 1] |
Character table: |
2 1 . . . 1 1 1 1 1 1 1 1 1 1 1 1 1 7 1 . . . 1 1 1 1 1 1 1 1 1 1 1 1 1 43 1 1 1 1 . . . . . . . . . . . . . 1a 43a 43b 43c 7a 7b 7c 7d 7e 7f 14a 14b 2a 14c 14d 14e 14f 2P 1a 43a 43b 43c 7c 7d 7e 7f 7a 7b 7a 7b 1a 7f 7d 7c 7e 3P 1a 43b 43c 43a 7b 7c 7d 7e 7f 7a 14b 14e 2a 14a 14f 14d 14c 5P 1a 43b 43c 43a 7f 7a 7b 7c 7d 7e 14c 14a 2a 14f 14e 14b 14d 7P 1a 43c 43a 43b 1a 1a 1a 1a 1a 1a 2a 2a 2a 2a 2a 2a 2a 11P 1a 43a 43b 43c 7e 7f 7a 7b 7c 7d 14f 14c 2a 14d 14b 14a 14e 13P 1a 43c 43a 43b 7d 7e 7f 7a 7b 7c 14d 14f 2a 14e 14a 14c 14b 17P 1a 43c 43a 43b 7b 7c 7d 7e 7f 7a 14b 14e 2a 14a 14f 14d 14c 19P 1a 43b 43c 43a 7f 7a 7b 7c 7d 7e 14c 14a 2a 14f 14e 14b 14d 23P 1a 43b 43c 43a 7c 7d 7e 7f 7a 7b 14e 14d 2a 14b 14c 14f 14a 29P 1a 43c 43a 43b 7a 7b 7c 7d 7e 7f 14a 14b 2a 14c 14d 14e 14f 31P 1a 43b 43c 43a 7b 7c 7d 7e 7f 7a 14b 14e 2a 14a 14f 14d 14c 37P 1a 43b 43c 43a 7c 7d 7e 7f 7a 7b 14e 14d 2a 14b 14c 14f 14a 41P 1a 43a 43b 43c 7d 7e 7f 7a 7b 7c 14d 14f 2a 14e 14a 14c 14b 43P 1a 1a 1a 1a 7a 7b 7c 7d 7e 7f 14a 14b 2a 14c 14d 14e 14f X.1 1 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 -1 -1 X.3 1 1 1 1 D F E /D /F /E -/F -/E -1 -/D -F -D -E X.4 1 1 1 1 E /D /F /E D F -D -F -1 -/E -/D -E -/F X.5 1 1 1 1 F E /D /F /E D -/E -D -1 -/F -E -F -/D X.6 1 1 1 1 /F /E D F E /D -E -/D -1 -F -/E -/F -D X.7 1 1 1 1 /E D F E /D /F -/D -/F -1 -E -D -/E -F X.8 1 1 1 1 /D /F /E D F E -F -E -1 -D -/F -/D -/E X.9 1 1 1 1 D F E /D /F /E /F /E 1 /D F D E X.10 1 1 1 1 E /D /F /E D F D F 1 /E /D E /F X.11 1 1 1 1 F E /D /F /E D /E D 1 /F E F /D X.12 1 1 1 1 /F /E D F E /D E /D 1 F /E /F D X.13 1 1 1 1 /E D F E /D /F /D /F 1 E D /E F X.14 1 1 1 1 /D /F /E D F E F E 1 D /F /D /E X.15 14 A B C . . . . . . . . . . . . . X.16 14 B C A . . . . . . . . . . . . . X.17 14 C A B . . . . . . . . . . . . . A = E(43)^3+E(43)^5+E(43)^6+E(43)^10+E(43)^12+E(43)^19+E(43)^20+E(43)^23+E(43)^24+E(43)^31+E(43)^33+E(43)^37+E(43)^38+E(43)^40 B = E(43)^7+E(43)^9+E(43)^13+E(43)^14+E(43)^15+E(43)^17+E(43)^18+E(43)^25+E(43)^26+E(43)^28+E(43)^29+E(43)^30+E(43)^34+E(43)^36 C = E(43)+E(43)^2+E(43)^4+E(43)^8+E(43)^11+E(43)^16+E(43)^21+E(43)^22+E(43)^27+E(43)^32+E(43)^35+E(43)^39+E(43)^41+E(43)^42 D = E(7)^6 E = E(7)^5 F = E(7)^4 |