Group action invariants
| Degree $n$ : | $29$ | |
| Transitive number $t$ : | $5$ | |
| Group : | $C_{29}:C_{14}$ | |
| Parity: | $1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (2,4,6,8,10,12,14,16,18,20,22,24,26,28)(3,5,7,9,11,13,15,17,19,21,23,25,27,29), (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) 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 $ | $1$ | $1$ | $()$ |
| $ 14, 14, 1 $ | $29$ | $14$ | $( 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28)( 3, 5, 7, 9,11,13,15,17,19,21,23, 25,27,29)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27)( 4, 8,12,16,20,24,28) ( 5, 9,13,17,21,25,29)$ |
| $ 14, 14, 1 $ | $29$ | $14$ | $( 2, 8,14,20,26, 4,10,16,22,28, 6,12,18,24)( 3, 9,15,21,27, 5,11,17,23,29, 7, 13,19,25)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23)( 4,12,20,28, 8,16,24) ( 5,13,21,29, 9,17,25)$ |
| $ 14, 14, 1 $ | $29$ | $14$ | $( 2,12,22, 4,14,24, 6,16,26, 8,18,28,10,20)( 3,13,23, 5,15,25, 7,17,27, 9,19, 29,11,21)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19)( 4,16,28,12,24, 8,20) ( 5,17,29,13,25, 9,21)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $29$ | $2$ | $( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25)(12,26) (13,27)(14,28)(15,29)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,18, 6,22,10,26,14)( 3,19, 7,23,11,27,15)( 4,20, 8,24,12,28,16) ( 5,21, 9,25,13,29,17)$ |
| $ 14, 14, 1 $ | $29$ | $14$ | $( 2,20,10,28,18, 8,26,16, 6,24,14, 4,22,12)( 3,21,11,29,19, 9,27,17, 7,25,15, 5,23,13)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,22,14, 6,26,18,10)( 3,23,15, 7,27,19,11)( 4,24,16, 8,28,20,12) ( 5,25,17, 9,29,21,13)$ |
| $ 14, 14, 1 $ | $29$ | $14$ | $( 2,24,18,12, 6,28,22,16,10, 4,26,20,14, 8)( 3,25,19,13, 7,29,23,17,11, 5,27, 21,15, 9)$ |
| $ 7, 7, 7, 7, 1 $ | $29$ | $7$ | $( 2,26,22,18,14,10, 6)( 3,27,23,19,15,11, 7)( 4,28,24,20,16,12, 8) ( 5,29,25,21,17,13, 9)$ |
| $ 14, 14, 1 $ | $29$ | $14$ | $( 2,28,26,24,22,20,18,16,14,12,10, 8, 6, 4)( 3,29,27,25,23,21,19,17,15,13,11, 9, 7, 5)$ |
| $ 29 $ | $14$ | $29$ | $( 1, 2, 3, 7, 4,24, 8,14, 5,12,25,27, 9,20,15,29, 6,23,13,11,26,19,28,22,10, 18,21,17,16)$ |
| $ 29 $ | $14$ | $29$ | $( 1, 3, 4, 8, 5,25, 9,15, 6,13,26,28,10,21,16, 2, 7,24,14,12,27,20,29,23,11, 19,22,18,17)$ |
Group invariants
| Order: | $406=2 \cdot 7 \cdot 29$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [406, 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 . .
29 1 . . . . . . . . . . . . . 1 1
1a 14a 7a 14b 7b 14c 7c 2a 7d 14d 7e 14e 7f 14f 29a 29b
2P 1a 7a 7b 7c 7d 7e 7f 1a 7a 7b 7c 7d 7e 7f 29b 29a
3P 1a 14b 7c 14d 7f 14a 7b 2a 7e 14f 7a 14c 7d 14e 29b 29a
5P 1a 14c 7e 14a 7c 14e 7a 2a 7f 14b 7d 14f 7b 14d 29a 29b
7P 1a 2a 1a 2a 1a 2a 1a 2a 1a 2a 1a 2a 1a 2a 29a 29b
11P 1a 14e 7d 14c 7a 14f 7e 2a 7b 14a 7f 14d 7c 14b 29b 29a
13P 1a 14f 7f 14e 7e 14d 7d 2a 7c 14c 7b 14b 7a 14a 29a 29b
17P 1a 14b 7c 14d 7f 14a 7b 2a 7e 14f 7a 14c 7d 14e 29b 29a
19P 1a 14c 7e 14a 7c 14e 7a 2a 7f 14b 7d 14f 7b 14d 29b 29a
23P 1a 14d 7b 14f 7d 14b 7f 2a 7a 14e 7c 14a 7e 14c 29a 29b
29P 1a 14a 7a 14b 7b 14c 7c 2a 7d 14d 7e 14e 7f 14f 1a 1a
X.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
X.3 1 A -B C -/C /B -/A -1 -A B -C /C -/B /A 1 1
X.4 1 B -/C /A -A C -/B -1 -B /C -/A A -C /B 1 1
X.5 1 C -/A B -/B A -/C -1 -C /A -B /B -A /C 1 1
X.6 1 /C -A /B -B /A -C -1 -/C A -/B B -/A C 1 1
X.7 1 /B -C A -/A /C -B -1 -/B C -A /A -/C B 1 1
X.8 1 /A -/B /C -C B -A -1 -/A /B -/C C -B A 1 1
X.9 1 -/A -/B -/C -C -B -A 1 -/A -/B -/C -C -B -A 1 1
X.10 1 -/B -C -A -/A -/C -B 1 -/B -C -A -/A -/C -B 1 1
X.11 1 -/C -A -/B -B -/A -C 1 -/C -A -/B -B -/A -C 1 1
X.12 1 -C -/A -B -/B -A -/C 1 -C -/A -B -/B -A -/C 1 1
X.13 1 -B -/C -/A -A -C -/B 1 -B -/C -/A -A -C -/B 1 1
X.14 1 -A -B -C -/C -/B -/A 1 -A -B -C -/C -/B -/A 1 1
X.15 14 . . . . . . . . . . . . . D *D
X.16 14 . . . . . . . . . . . . . *D D
A = -E(7)
B = -E(7)^2
C = -E(7)^3
D = E(29)^2+E(29)^3+E(29)^8+E(29)^10+E(29)^11+E(29)^12+E(29)^14+E(29)^15+E(29)^17+E(29)^18+E(29)^19+E(29)^21+E(29)^26+E(29)^27
= (-1-Sqrt(29))/2 = -1-b29
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