Group action invariants
| Degree $n$ : | $43$ | |
| Transitive number $t$ : | $6$ | |
| Group : | $C_{43}:C_{14}$ | |
| Parity: | $-1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| 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) | |
| $|\Aut(F/K)|$: | $1$ |
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
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