Group action invariants
| Degree $n$ : | $29$ | |
| Transitive number $t$ : | $4$ | |
| Group : | $C_{29}:C_{7}$ | |
| Parity: | $1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (2,6,10,14,18,22,26)(3,7,11,15,19,23,27)(4,8,12,16,20,24,28)(5,9,13,17,21,25,29), (1,2,3,7,4,24,8,14,5,12,25,27,9,20,15,29,6,23,13,11,26,19,28,22,10,18,21,17,16) | |
| $|\Aut(F/K)|$: | $1$ |
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 $ | $1$ | $1$ | $()$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27)( 4, 8,12,16,20,24,28) ( 5, 9,13,17,21,25,29)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23)( 4,12,20,28, 8,16,24) ( 5,13,21,29, 9,17,25)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19)( 4,16,28,12,24, 8,20) ( 5,17,29,13,25, 9,21)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,18, 6,22,10,26,14)( 3,19, 7,23,11,27,15)( 4,20, 8,24,12,28,16) ( 5,21, 9,25,13,29,17)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,22,14, 6,26,18,10)( 3,23,15, 7,27,19,11)( 4,24,16, 8,28,20,12) ( 5,25,17, 9,29,21,13)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,26,22,18,14,10, 6)( 3,27,23,19,15,11, 7)( 4,28,24,20,16,12, 8) ( 5,29,25,21,17,13, 9)$ |
| $ 29 $ | $7$ | $29$ | $( 1, 2, 3, 7, 4,24, 8,14, 5,12,25,27, 9,20,15,29, 6,23,13,11,26,19,28,22,10, 18,21,17,16)$ |
| $ 29 $ | $7$ | $29$ | $( 1, 3, 4, 8, 5,25, 9,15, 6,13,26,28,10,21,16, 2, 7,24,14,12,27,20,29,23,11, 19,22,18,17)$ |
| $ 29 $ | $7$ | $29$ | $( 1, 4, 5, 9, 6,26,10,16, 7,14,27,29,11,22,17, 3, 8,25,15,13,28,21, 2,24,12, 20,23,19,18)$ |
| $ 29 $ | $7$ | $29$ | $( 1, 5, 6,10, 7,27,11,17, 8,15,28, 2,12,23,18, 4, 9,26,16,14,29,22, 3,25,13, 21,24,20,19)$ |
Group invariants
| Order: | $203=7 \cdot 29$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [203, 1] |
| Character table: |
7 1 1 1 1 1 1 1 . . . .
29 1 . . . . . . 1 1 1 1
1a 7a 7b 7c 7d 7e 7f 29a 29b 29c 29d
2P 1a 7b 7d 7f 7a 7c 7e 29b 29c 29d 29a
3P 1a 7c 7f 7b 7e 7a 7d 29b 29c 29d 29a
5P 1a 7e 7c 7a 7f 7d 7b 29c 29d 29a 29b
7P 1a 1a 1a 1a 1a 1a 1a 29a 29b 29c 29d
11P 1a 7d 7a 7e 7b 7f 7c 29b 29c 29d 29a
13P 1a 7f 7e 7d 7c 7b 7a 29c 29d 29a 29b
17P 1a 7c 7f 7b 7e 7a 7d 29b 29c 29d 29a
19P 1a 7e 7c 7a 7f 7d 7b 29b 29c 29d 29a
23P 1a 7b 7d 7f 7a 7c 7e 29a 29b 29c 29d
29P 1a 7a 7b 7c 7d 7e 7f 1a 1a 1a 1a
X.1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 A B C /C /B /A 1 1 1 1
X.3 1 B /C /A A C /B 1 1 1 1
X.4 1 C /A B /B A /C 1 1 1 1
X.5 1 /C A /B B /A C 1 1 1 1
X.6 1 /B C A /A /C B 1 1 1 1
X.7 1 /A /B /C C B A 1 1 1 1
X.8 7 . . . . . . D E /D /E
X.9 7 . . . . . . E /D /E D
X.10 7 . . . . . . /E D E /D
X.11 7 . . . . . . /D /E D E
A = E(7)^6
B = E(7)^5
C = E(7)^4
D = E(29)^4+E(29)^5+E(29)^6+E(29)^9+E(29)^13+E(29)^22+E(29)^28
E = E(29)^8+E(29)^10+E(29)^12+E(29)^15+E(29)^18+E(29)^26+E(29)^27
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