Group action invariants
| Degree $n$ : | $43$ | |
| Transitive number $t$ : | $5$ | |
| Group : | $C_{43}:C_{7}$ | |
| Parity: | $1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43), (1,41,4,35,16,11,21)(2,39,8,27,32,22,42)(3,37,12,19,5,33,20)(6,31,24,38,10,23,40)(7,29,28,30,26,34,18)(9,25,36,14,15,13,17) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 7: $C_7$ 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 $ | $1$ | $1$ | $()$ |
| $ 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 $ | $43$ | $7$ | $( 2,12,36,42,22,17, 5)( 3,23,28,40,43,33, 9)( 4,34,20,38,21, 6,13) ( 7,24,39,32,41,11,25)( 8,35,31,30,19,27,29)(10,14,15,26,18,16,37)$ |
| $ 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 $ | $43$ | $7$ | $( 2,22,12,17,36, 5,42)( 3,43,23,33,28, 9,40)( 4,21,34, 6,20,13,38) ( 7,41,24,11,39,25,32)( 8,19,35,27,31,29,30)(10,18,14,16,15,37,26)$ |
| $ 7, 7, 7, 7, 7, 7, 1 $ | $43$ | $7$ | $( 2,36,22, 5,12,42,17)( 3,28,43, 9,23,40,33)( 4,20,21,13,34,38, 6) ( 7,39,41,25,24,32,11)( 8,31,19,29,35,30,27)(10,15,18,37,14,26,16)$ |
| $ 7, 7, 7, 7, 7, 7, 1 $ | $43$ | $7$ | $( 2,42, 5,36,17,12,22)( 3,40, 9,28,33,23,43)( 4,38,13,20, 6,34,21) ( 7,32,25,39,11,24,41)( 8,30,29,31,27,35,19)(10,26,37,15,16,14,18)$ |
| $ 43 $ | $7$ | $43$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)$ |
| $ 43 $ | $7$ | $43$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42)$ |
| $ 43 $ | $7$ | $43$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40,43, 3, 6, 9,12,15,18,21,24,27,30, 33,36,39,42, 2, 5, 8,11,14,17,20,23,26,29,32,35,38,41)$ |
| $ 43 $ | $7$ | $43$ | $( 1, 7,13,19,25,31,37,43, 6,12,18,24,30,36,42, 5,11,17,23,29,35,41, 4,10,16, 22,28,34,40, 3, 9,15,21,27,33,39, 2, 8,14,20,26,32,38)$ |
| $ 43 $ | $7$ | $43$ | $( 1, 8,15,22,29,36,43, 7,14,21,28,35,42, 6,13,20,27,34,41, 5,12,19,26,33,40, 4,11,18,25,32,39, 3,10,17,24,31,38, 2, 9,16,23,30,37)$ |
| $ 43 $ | $7$ | $43$ | $( 1,10,19,28,37, 3,12,21,30,39, 5,14,23,32,41, 7,16,25,34,43, 9,18,27,36, 2, 11,20,29,38, 4,13,22,31,40, 6,15,24,33,42, 8,17,26,35)$ |
Group invariants
| Order: | $301=7 \cdot 43$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [301, 1] |
| Character table: |
7 1 1 1 1 1 1 1 . . . . . .
43 1 . . . . . . 1 1 1 1 1 1
1a 7a 7b 7c 7d 7e 7f 43a 43b 43c 43d 43e 43f
2P 1a 7c 7e 7f 7b 7d 7a 43b 43a 43d 43c 43f 43e
3P 1a 7d 7f 7b 7c 7a 7e 43c 43d 43f 43e 43a 43b
5P 1a 7e 7c 7d 7a 7f 7b 43c 43d 43f 43e 43a 43b
7P 1a 1a 1a 1a 1a 1a 1a 43e 43f 43a 43b 43d 43c
11P 1a 7f 7d 7a 7e 7b 7c 43a 43b 43c 43d 43e 43f
13P 1a 7b 7a 7e 7f 7c 7d 43f 43e 43b 43a 43c 43d
17P 1a 7d 7f 7b 7c 7a 7e 43f 43e 43b 43a 43c 43d
19P 1a 7e 7c 7d 7a 7f 7b 43c 43d 43f 43e 43a 43b
23P 1a 7c 7e 7f 7b 7d 7a 43d 43c 43e 43f 43b 43a
29P 1a 7a 7b 7c 7d 7e 7f 43e 43f 43a 43b 43d 43c
31P 1a 7d 7f 7b 7c 7a 7e 43d 43c 43e 43f 43b 43a
37P 1a 7c 7e 7f 7b 7d 7a 43c 43d 43f 43e 43a 43b
41P 1a 7b 7a 7e 7f 7c 7d 43a 43b 43c 43d 43e 43f
43P 1a 7a 7b 7c 7d 7e 7f 1a 1a 1a 1a 1a 1a
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 A /A B C /B /C 1 1 1 1 1 1
X.3 1 B /B /C /A C A 1 1 1 1 1 1
X.4 1 C /C /A B A /B 1 1 1 1 1 1
X.5 1 /C C A /B /A B 1 1 1 1 1 1
X.6 1 /B B C A /C /A 1 1 1 1 1 1
X.7 1 /A A /B /C B C 1 1 1 1 1 1
X.8 7 . . . . . . D /D /E E /F F
X.9 7 . . . . . . E /E /F F /D D
X.10 7 . . . . . . F /F /D D /E E
X.11 7 . . . . . . /E E F /F D /D
X.12 7 . . . . . . /F F D /D E /E
X.13 7 . . . . . . /D D E /E F /F
A = E(7)^6
B = E(7)^5
C = E(7)^4
D = E(43)^2+E(43)^8+E(43)^22+E(43)^27+E(43)^32+E(43)^39+E(43)^42
E = E(43)^3+E(43)^5+E(43)^12+E(43)^19+E(43)^20+E(43)^33+E(43)^37
F = E(43)^7+E(43)^18+E(43)^26+E(43)^28+E(43)^29+E(43)^30+E(43)^34
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