Group action invariants
| Degree $n$ : | $29$ | |
| Transitive number $t$ : | $3$ | |
| Group : | $C_{29}:C_{4}$ | |
| Parity: | $-1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(8,15,22,29), (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: $C_2$ 4: $C_4$ 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 $ | $1$ | $1$ | $()$ |
| $ 4, 4, 4, 4, 4, 4, 4, 1 $ | $29$ | $4$ | $( 2, 9,16,23)( 3,10,17,24)( 4,11,18,25)( 5,12,19,26)( 6,13,20,27)( 7,14,21,28) ( 8,15,22,29)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $29$ | $2$ | $( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25)(12,26) (13,27)(14,28)(15,29)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 1 $ | $29$ | $4$ | $( 2,23,16, 9)( 3,24,17,10)( 4,25,18,11)( 5,26,19,12)( 6,27,20,13)( 7,28,21,14) ( 8,29,22,15)$ |
| $ 29 $ | $4$ | $29$ | $( 1, 2, 3, 7, 4,24, 8,14, 5,12,25,27, 9,20,15,29, 6,23,13,11,26,19,28,22,10, 18,21,17,16)$ |
| $ 29 $ | $4$ | $29$ | $( 1, 3, 4, 8, 5,25, 9,15, 6,13,26,28,10,21,16, 2, 7,24,14,12,27,20,29,23,11, 19,22,18,17)$ |
| $ 29 $ | $4$ | $29$ | $( 1, 4, 5, 9, 6,26,10,16, 7,14,27,29,11,22,17, 3, 8,25,15,13,28,21, 2,24,12, 20,23,19,18)$ |
| $ 29 $ | $4$ | $29$ | $( 1, 5, 6,10, 7,27,11,17, 8,15,28, 2,12,23,18, 4, 9,26,16,14,29,22, 3,25,13, 21,24,20,19)$ |
| $ 29 $ | $4$ | $29$ | $( 1, 6, 7,11, 8,28,12,18, 9,16,29, 3,13,24,19, 5,10,27,17,15, 2,23, 4,26,14, 22,25,21,20)$ |
| $ 29 $ | $4$ | $29$ | $( 1, 7, 8,12, 9,29,13,19,10,17, 2, 4,14,25,20, 6,11,28,18,16, 3,24, 5,27,15, 23,26,22,21)$ |
| $ 29 $ | $4$ | $29$ | $( 1, 8, 9,13,10, 2,14,20,11,18, 3, 5,15,26,21, 7,12,29,19,17, 4,25, 6,28,16, 24,27,23,22)$ |
Group invariants
| Order: | $116=2^{2} \cdot 29$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [116, 3] |
| Character table: |
2 2 2 2 2 . . . . . . .
29 1 . . . 1 1 1 1 1 1 1
1a 4a 2a 4b 29a 29b 29c 29d 29e 29f 29g
2P 1a 2a 1a 2a 29b 29c 29d 29e 29f 29g 29a
3P 1a 4b 2a 4a 29f 29g 29a 29b 29c 29d 29e
5P 1a 4a 2a 4b 29b 29c 29d 29e 29f 29g 29a
7P 1a 4b 2a 4a 29f 29g 29a 29b 29c 29d 29e
11P 1a 4b 2a 4a 29e 29f 29g 29a 29b 29c 29d
13P 1a 4a 2a 4b 29e 29f 29g 29a 29b 29c 29d
17P 1a 4a 2a 4b 29a 29b 29c 29d 29e 29f 29g
19P 1a 4b 2a 4a 29c 29d 29e 29f 29g 29a 29b
23P 1a 4b 2a 4a 29g 29a 29b 29c 29d 29e 29f
29P 1a 4a 2a 4b 1a 1a 1a 1a 1a 1a 1a
X.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
X.3 1 A -1 -A 1 1 1 1 1 1 1
X.4 1 -A -1 A 1 1 1 1 1 1 1
X.5 4 . . . B G H D F C E
X.6 4 . . . C E B G H D F
X.7 4 . . . D F C E B G H
X.8 4 . . . E B G H D F C
X.9 4 . . . F C E B G H D
X.10 4 . . . G H D F C E B
X.11 4 . . . H D F C E B G
A = -E(4)
= -Sqrt(-1) = -i
B = E(29)^11+E(29)^13+E(29)^16+E(29)^18
C = E(29)^4+E(29)^10+E(29)^19+E(29)^25
D = E(29)+E(29)^12+E(29)^17+E(29)^28
E = E(29)^8+E(29)^9+E(29)^20+E(29)^21
F = E(29)^2+E(29)^5+E(29)^24+E(29)^27
G = E(29)^3+E(29)^7+E(29)^22+E(29)^26
H = E(29)^6+E(29)^14+E(29)^15+E(29)^23
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