Group action invariants
| Degree $n$ : | $17$ | |
| Transitive number $t$ : | $7$ | |
| Group : | $\PSL(2,16):C_2$ | |
| Parity: | $1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (2,3)(4,9)(5,7)(6,8)(10,14)(11,13)(12,15)(16,17), (1,16)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15), (1,6,13,5,4,2,15,10,14,12,3,9,7,11,8), (1,7)(2,13)(3,10)(4,11)(5,12)(8,14) | |
| $|\Aut(F/K)|$: | $1$ |
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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 1, 1 $ | $272$ | $3$ | $( 1,14, 3)( 2,12, 4)( 5,11,13)( 7, 8,10)( 9,17,16)$ |
| $ 5, 5, 5, 1, 1 $ | $272$ | $5$ | $( 1,16, 7, 4,11)( 2,13,14, 9, 8)( 3,17,10,12, 5)$ |
| $ 5, 5, 5, 1, 1 $ | $272$ | $5$ | $( 1, 4,16,11, 7)( 2, 9,13, 8,14)( 3,12,17, 5,10)$ |
| $ 15, 1, 1 $ | $544$ | $15$ | $( 1,10,13,16,12,14, 7, 5, 9, 4, 3, 8,11,17, 2)$ |
| $ 15, 1, 1 $ | $544$ | $15$ | $( 1, 5, 2, 7,17,14,11,12, 8,16, 3,13, 4,10, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $68$ | $2$ | $( 1,14)( 2, 4)( 6,15)( 7, 8)( 9,16)(11,13)$ |
| $ 10, 5, 2 $ | $816$ | $10$ | $( 1, 8,11, 9, 4,14, 7,13,16, 2)( 3,10, 5,17,12)( 6,15)$ |
| $ 10, 5, 2 $ | $816$ | $10$ | $( 1,13, 4, 8,16,14,11, 2, 7, 9)( 3, 5,12,10,17)( 6,15)$ |
| $ 6, 6, 3, 1, 1 $ | $1360$ | $6$ | $( 1, 8, 3, 7,14,10)( 2, 5, 4,13,12,11)( 9,17,16)$ |
| $ 17 $ | $480$ | $17$ | $( 1,13,12, 2,17,10, 4, 5, 9, 6,15, 8,11, 3,16, 7,14)$ |
| $ 17 $ | $480$ | $17$ | $( 1,12,17, 4, 9,15,11,16,14,13, 2,10, 5, 6, 8, 3, 7)$ |
| $ 17 $ | $480$ | $17$ | $( 1, 4,11,13, 5, 3,12, 9,16, 2, 6, 7,17,15,14,10, 8)$ |
| $ 17 $ | $480$ | $17$ | $( 1,11, 5,12,16, 6,17,14, 8, 4,13, 3, 9, 2, 7,15,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $255$ | $2$ | $( 1, 2)( 3, 6)( 4,11)( 5,15)( 7,17)( 8,12)( 9,14)(13,16)$ |
| $ 4, 4, 4, 4, 1 $ | $1020$ | $4$ | $( 1,11, 2, 4)( 3, 8, 6,12)( 5,16,15,13)( 7, 9,17,14)$ |
Group invariants
| Order: | $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: |
2 5 . . . . 3 1 1 1 1 1 . . 5 3 1
3 1 . . . . 1 1 1 . . 1 1 1 . . 1
5 1 . . . . 1 1 1 1 1 1 1 1 . . .
17 1 1 1 1 1 . . . . . . . . . . .
1a 17a 17b 17c 17d 2a 5a 5b 10a 10b 3a 15a 15b 2b 4a 6a
2P 1a 17b 17a 17d 17c 1a 5b 5a 5a 5b 3a 15b 15a 1a 2b 3a
3P 1a 17d 17c 17a 17b 2a 5b 5a 10b 10a 1a 5a 5b 2b 4a 2a
5P 1a 17d 17c 17a 17b 2a 1a 1a 2a 2a 3a 3a 3a 2b 4a 6a
7P 1a 17c 17d 17b 17a 2a 5b 5a 10b 10a 3a 15b 15a 2b 4a 6a
11P 1a 17c 17d 17b 17a 2a 5a 5b 10a 10b 3a 15a 15b 2b 4a 6a
13P 1a 17a 17b 17c 17d 2a 5b 5a 10b 10a 3a 15b 15a 2b 4a 6a
17P 1a 1a 1a 1a 1a 2a 5b 5a 10b 10a 3a 15b 15a 2b 4a 6a
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 16 -1 -1 -1 -1 -4 1 1 1 1 1 1 1 . . -1
X.4 16 -1 -1 -1 -1 4 1 1 -1 -1 1 1 1 . . 1
X.5 17 . . . . 5 2 2 . . -1 -1 -1 1 1 -1
X.6 17 . . . . -5 2 2 . . -1 -1 -1 1 -1 1
X.7 17 . . . . -3 E *E *E E 2 *E E 1 1 .
X.8 17 . . . . -3 *E E E *E 2 E *E 1 1 .
X.9 17 . . . . 3 E *E -*E -E 2 *E E 1 -1 .
X.10 17 . . . . 3 *E E -E -*E 2 E *E 1 -1 .
X.11 30 A B D C . . . . . . . . -2 . .
X.12 30 B A C D . . . . . . . . -2 . .
X.13 30 C D A B . . . . . . . . -2 . .
X.14 30 D C B A . . . . . . . . -2 . .
X.15 34 . . . . . F *F . . -2 -*E -E 2 . .
X.16 34 . . . . . *F F . . -2 -E -*E 2 . .
A = -E(17)^6-E(17)^7-E(17)^10-E(17)^11
B = -E(17)^3-E(17)^5-E(17)^12-E(17)^14
C = -E(17)-E(17)^4-E(17)^13-E(17)^16
D = -E(17)^2-E(17)^8-E(17)^9-E(17)^15
E = E(5)^2+E(5)^3
= (-1-Sqrt(5))/2 = -1-b5
F = 2*E(5)^2+2*E(5)^3
= -1-Sqrt(5) = -1-r5
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