Group action invariants
| Degree $n$ : | $17$ | |
| Transitive number $t$ : | $4$ | |
| Group : | $C_{17}:C_{8}$ | |
| Parity: | $1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4: $C_4$ 8: $C_8$ 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 $ | $1$ | $1$ | $()$ |
| $ 8, 8, 1 $ | $17$ | $8$ | $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)$ |
| $ 4, 4, 4, 4, 1 $ | $17$ | $4$ | $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)$ |
| $ 8, 8, 1 $ | $17$ | $8$ | $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)$ |
| $ 8, 8, 1 $ | $17$ | $8$ | $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)$ |
| $ 4, 4, 4, 4, 1 $ | $17$ | $4$ | $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)$ |
| $ 8, 8, 1 $ | $17$ | $8$ | $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)$ |
| $ 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 $ | $8$ | $17$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$ |
| $ 17 $ | $8$ | $17$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$ |
Group invariants
| Order: | $136=2^{3} \cdot 17$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [136, 12] |
| Character table: |
2 3 3 3 3 3 3 3 3 . .
17 1 . . . . . . . 1 1
1a 8a 4a 8b 8c 4b 8d 2a 17a 17b
2P 1a 4a 2a 4b 4b 2a 4a 1a 17a 17b
3P 1a 8b 4b 8a 8d 4a 8c 2a 17b 17a
5P 1a 8d 4a 8c 8b 4b 8a 2a 17b 17a
7P 1a 8c 4b 8d 8a 4a 8b 2a 17b 17a
11P 1a 8b 4b 8a 8d 4a 8c 2a 17b 17a
13P 1a 8d 4a 8c 8b 4b 8a 2a 17a 17b
17P 1a 8a 4a 8b 8c 4b 8d 2a 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 1 A -1 -A -A -1 A 1 1 1
X.4 1 -A -1 A A -1 -A 1 1 1
X.5 1 B -A -/B /B A -B -1 1 1
X.6 1 -/B A B -B -A /B -1 1 1
X.7 1 /B A -B B -A -/B -1 1 1
X.8 1 -B -A /B -/B A B -1 1 1
X.9 8 . . . . . . . C *C
X.10 8 . . . . . . . *C C
A = -E(4)
= -Sqrt(-1) = -i
B = -E(8)
C = E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12+E(17)^14
= (-1-Sqrt(17))/2 = -1-b17
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