Group action invariants
| Degree $n$: | $17$ | |
| Transitive number $t$: | $8$ | |
| Group: | $\PSL(2,16):C_4$ | |
| Parity: | $1$ | |
| Primitive: | yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (2,3)(4,9)(5,7)(6,8)(10,14)(11,13)(12,15)(16,17), (1,12,7,5)(2,4,13,11)(3,8,10,14)(6,9), (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) |
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $255$ | $2$ | $( 1,17)( 3,16)( 4,12)( 5, 7)( 6,15)( 8,13)( 9,14)(10,11)$ |
| $ 4, 4, 4, 4, 1 $ | $1020$ | $4$ | $( 1, 7,17, 5)( 3,11,16,10)( 4,13,12, 8)( 6, 9,15,14)$ |
| $ 8, 8, 1 $ | $2040$ | $8$ | $( 1,13, 5, 4,17, 8, 7,12)( 3, 9,10, 6,16,14,11,15)$ |
| $ 8, 8, 1 $ | $2040$ | $8$ | $( 1,12, 7, 8,17, 4, 5,13)( 3,15,11,14,16, 6,10, 9)$ |
| $ 17 $ | $960$ | $17$ | $( 1,16,13, 7, 3, 4,11, 9,15,17,14, 8,10, 5, 2, 6,12)$ |
| $ 17 $ | $960$ | $17$ | $( 1,11,10,16, 9, 5,13,15, 2, 7,17, 6, 3,14,12, 4, 8)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $68$ | $2$ | $( 1,13)( 2, 6)( 4,12)( 5, 9)( 8,16)(10,14)$ |
| $ 5, 5, 5, 1, 1 $ | $544$ | $5$ | $( 1, 8, 2,10,12)( 3,15,11,17, 7)( 4,13,16, 6,14)$ |
| $ 10, 5, 2 $ | $1632$ | $10$ | $( 1,14, 8, 4, 2,13,10,16,12, 6)( 3,17,15, 7,11)( 5, 9)$ |
| $ 3, 3, 3, 3, 3, 1, 1 $ | $272$ | $3$ | $( 1,17,12)( 2,16, 8)( 3, 4,10)( 5,13,15)( 7, 9,14)$ |
| $ 15, 1, 1 $ | $1088$ | $15$ | $( 1, 3,16, 5, 9,17, 4, 8,13,14,12,10, 2,15, 7)$ |
| $ 4, 4, 4, 2, 1, 1, 1 $ | $680$ | $4$ | $( 1, 5, 6,10)( 2, 4, 7, 8)( 9,11)(12,14,17,15)$ |
| $ 4, 4, 4, 2, 1, 1, 1 $ | $680$ | $4$ | $( 1,10, 6, 5)( 2, 8, 7, 4)( 9,11)(12,15,17,14)$ |
| $ 6, 6, 3, 1, 1 $ | $1360$ | $6$ | $( 1, 7,15, 6, 2,14)( 3,13,16)( 4,17, 5, 8,12,10)$ |
| $ 12, 3, 2 $ | $1360$ | $12$ | $( 1,17, 7, 5,15, 8, 6,12, 2,10,14, 4)( 3,16,13)( 9,11)$ |
| $ 12, 3, 2 $ | $1360$ | $12$ | $( 1,12, 7,10,15, 4, 6,17, 2, 5,14, 8)( 3,16,13)( 9,11)$ |
Group invariants
| Order: | $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | no | |
| GAP id: | not available |
| Character table: |
2 6 1 2 . 4 1 . . 6 4 3 3 3 3 2 2 2
3 1 1 1 1 1 . . . . . . . 1 1 1 1 1
5 1 1 1 1 1 1 . . . . . . . . . . .
17 1 . . . . . 1 1 . . . . . . . . .
1a 5a 3a 15a 2a 10a 17a 17b 2b 4a 8a 8b 4b 4c 6a 12a 12b
2P 1a 5a 3a 15a 1a 5a 17a 17b 1a 2b 4a 4a 2a 2a 3a 6a 6a
3P 1a 5a 1a 5a 2a 10a 17b 17a 2b 4a 8b 8a 4c 4b 2a 4b 4c
5P 1a 1a 3a 3a 2a 2a 17b 17a 2b 4a 8a 8b 4b 4c 6a 12a 12b
7P 1a 5a 3a 15a 2a 10a 17b 17a 2b 4a 8b 8a 4c 4b 6a 12b 12a
11P 1a 5a 3a 15a 2a 10a 17b 17a 2b 4a 8b 8a 4c 4b 6a 12b 12a
13P 1a 5a 3a 15a 2a 10a 17a 17b 2b 4a 8a 8b 4b 4c 6a 12a 12b
17P 1a 5a 3a 15a 2a 10a 1a 1a 2b 4a 8a 8b 4b 4c 6a 12a 12b
X.1 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 -1
X.3 1 1 1 1 -1 -1 1 1 1 -1 B -B -B B -1 B -B
X.4 1 1 1 1 -1 -1 1 1 1 -1 -B B B -B -1 -B B
X.5 16 1 1 1 4 -1 -1 -1 . . . . -2 -2 1 1 1
X.6 16 1 1 1 4 -1 -1 -1 . . . . 2 2 1 -1 -1
X.7 16 1 1 1 -4 1 -1 -1 . . . . C -C -1 B -B
X.8 16 1 1 1 -4 1 -1 -1 . . . . -C C -1 -B B
X.9 17 2 -1 -1 5 . . . 1 1 -1 -1 1 1 -1 1 1
X.10 17 2 -1 -1 5 . . . 1 1 1 1 -1 -1 -1 -1 -1
X.11 17 2 -1 -1 -5 . . . 1 -1 B -B B -B 1 -B B
X.12 17 2 -1 -1 -5 . . . 1 -1 -B B -B B 1 B -B
X.13 34 -1 4 -1 -6 -1 . . 2 2 . . . . . . .
X.14 34 -1 4 -1 6 1 . . 2 -2 . . . . . . .
X.15 60 . . . . . A *A -4 . . . . . . . .
X.16 60 . . . . . *A A -4 . . . . . . . .
X.17 68 -2 -4 1 . . . . 4 . . . . . . . .
A = -E(17)-E(17)^2-E(17)^4-E(17)^8-E(17)^9-E(17)^13-E(17)^15-E(17)^16
= (1-Sqrt(17))/2 = -b17
B = -E(4)
= -Sqrt(-1) = -i
C = -2*E(4)
= -2*Sqrt(-1) = -2i
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