Group action invariants
| Degree $n$ : | $17$ | |
| Transitive number $t$ : | $2$ | |
| Group : | $D_{17}$ | |
| 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), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $17$ | $2$ | $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$ |
| $ 17 $ | $2$ | $17$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$ |
| $ 17 $ | $2$ | $17$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)$ |
| $ 17 $ | $2$ | $17$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$ |
| $ 17 $ | $2$ | $17$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)$ |
| $ 17 $ | $2$ | $17$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)$ |
| $ 17 $ | $2$ | $17$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)$ |
| $ 17 $ | $2$ | $17$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)$ |
| $ 17 $ | $2$ | $17$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)$ |
Group invariants
| Order: | $34=2 \cdot 17$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [34, 1] |
| Character table: |
2 1 1 . . . . . . . .
17 1 . 1 1 1 1 1 1 1 1
1a 2a 17a 17b 17c 17d 17e 17f 17g 17h
2P 1a 1a 17b 17d 17f 17h 17g 17e 17c 17a
3P 1a 2a 17c 17f 17h 17e 17b 17a 17d 17g
5P 1a 2a 17e 17g 17b 17c 17h 17d 17a 17f
7P 1a 2a 17g 17c 17d 17f 17a 17h 17b 17e
11P 1a 2a 17f 17e 17a 17g 17d 17b 17h 17c
13P 1a 2a 17d 17h 17e 17a 17c 17g 17f 17b
17P 1a 2a 1a 1a 1a 1a 1a 1a 1a 1a
X.1 1 1 1 1 1 1 1 1 1 1
X.2 1 -1 1 1 1 1 1 1 1 1
X.3 2 . A E D B F G H C
X.4 2 . B C F A D H G E
X.5 2 . C A H E G D F B
X.6 2 . D G C F E A B H
X.7 2 . E B G C H F D A
X.8 2 . F H E D C B A G
X.9 2 . G F A H B E C D
X.10 2 . H D B G A C E F
A = E(17)^5+E(17)^12
B = E(17)^3+E(17)^14
C = E(17)^6+E(17)^11
D = E(17)^2+E(17)^15
E = E(17)^7+E(17)^10
F = E(17)^8+E(17)^9
G = E(17)^4+E(17)^13
H = E(17)+E(17)^16
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