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) | |
| 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) | |
| $|\Aut(F/K)|$: | $1$ |
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $68$ | $2$ | $( 1, 9)( 3, 7)( 4, 8)( 5,13)(11,15)(12,16)$ |
| $ 3, 3, 3, 3, 3, 1, 1 $ | $272$ | $3$ | $( 1,15, 3)( 2,17,10)( 4,13,16)( 5,12, 8)( 7, 9,11)$ |
| $ 6, 6, 3, 1, 1 $ | $1360$ | $6$ | $( 1, 7,15, 9, 3,11)( 2,10,17)( 4,12,13, 8,16, 5)$ |
| $ 4, 4, 4, 2, 1, 1, 1 $ | $680$ | $4$ | $( 1,13, 9, 5)( 3, 4, 7, 8)( 6,14)(11,12,15,16)$ |
| $ 4, 4, 4, 2, 1, 1, 1 $ | $680$ | $4$ | $( 1, 5, 9,13)( 3, 8, 7, 4)( 6,14)(11,16,15,12)$ |
| $ 12, 3, 2 $ | $1360$ | $12$ | $( 1, 8,11,13, 3,12, 9, 4,15, 5, 7,16)( 2,10,17)( 6,14)$ |
| $ 12, 3, 2 $ | $1360$ | $12$ | $( 1, 4,11, 5, 3,16, 9, 8,15,13, 7,12)( 2,10,17)( 6,14)$ |
| $ 5, 5, 5, 1, 1 $ | $544$ | $5$ | $( 1, 5,11, 8, 3)( 2,10,17, 6,14)( 4, 7, 9,13,15)$ |
| $ 10, 5, 2 $ | $1632$ | $10$ | $( 1, 4, 5, 7,11, 9, 8,13, 3,15)( 2, 6,10,14,17)(12,16)$ |
| $ 15, 1, 1 $ | $1088$ | $15$ | $( 1,15, 6, 5, 4,14,11, 7, 2, 8, 9,10, 3,13,17)$ |
| $ 17 $ | $960$ | $17$ | $( 1,17,10, 5,11,13, 9, 8, 4,12, 7,15, 3, 2, 6,16,14)$ |
| $ 17 $ | $960$ | $17$ | $( 1, 9, 3,17, 8, 2,10, 4, 6, 5,12,16,11, 7,14,13,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $255$ | $2$ | $( 1,10)( 2, 6)( 3,15)( 4, 9)( 5,12)( 7,17)( 8,16)(13,14)$ |
| $ 4, 4, 4, 4, 1 $ | $1020$ | $4$ | $( 1, 5,10,12)( 2,15, 6, 3)( 4,14, 9,13)( 7,16,17, 8)$ |
| $ 8, 8, 1 $ | $2040$ | $8$ | $( 1,17, 5, 8,10, 7,12,16)( 2,13,15, 4, 6,14, 3, 9)$ |
| $ 8, 8, 1 $ | $2040$ | $8$ | $( 1,16,12, 7,10, 8, 5,17)( 2, 9, 3,14, 6, 4,15,13)$ |
Group invariants
| Order: | $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: |
2 6 . . 6 4 3 3 2 1 . 4 3 3 2 2 2 1
3 1 . . . . . . 1 1 1 1 1 1 1 1 1 .
5 1 . . . . . . 1 1 1 1 . . . . . 1
17 1 1 1 . . . . . . . . . . . . . .
1a 17a 17b 2a 4a 8a 8b 3a 5a 15a 2b 4b 4c 6a 12a 12b 10a
2P 1a 17a 17b 1a 2a 4a 4a 3a 5a 15a 1a 2b 2b 3a 6a 6a 5a
3P 1a 17b 17a 2a 4a 8b 8a 1a 5a 5a 2b 4c 4b 2b 4b 4c 10a
5P 1a 17b 17a 2a 4a 8a 8b 3a 1a 3a 2b 4b 4c 6a 12a 12b 2b
7P 1a 17b 17a 2a 4a 8b 8a 3a 5a 15a 2b 4c 4b 6a 12b 12a 10a
11P 1a 17b 17a 2a 4a 8b 8a 3a 5a 15a 2b 4c 4b 6a 12b 12a 10a
13P 1a 17a 17b 2a 4a 8a 8b 3a 5a 15a 2b 4b 4c 6a 12a 12b 10a
17P 1a 1a 1a 2a 4a 8a 8b 3a 5a 15a 2b 4b 4c 6a 12a 12b 10a
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 B -B 1 1 1 -1 -B B -1 B -B -1
X.4 1 1 1 1 -1 -B B 1 1 1 -1 B -B -1 -B B -1
X.5 16 -1 -1 . . . . 1 1 1 4 -2 -2 1 1 1 -1
X.6 16 -1 -1 . . . . 1 1 1 4 2 2 1 -1 -1 -1
X.7 16 -1 -1 . . . . 1 1 1 -4 C -C -1 B -B 1
X.8 16 -1 -1 . . . . 1 1 1 -4 -C C -1 -B B 1
X.9 17 . . 1 1 -1 -1 -1 2 -1 5 1 1 -1 1 1 .
X.10 17 . . 1 1 1 1 -1 2 -1 5 -1 -1 -1 -1 -1 .
X.11 17 . . 1 -1 B -B -1 2 -1 -5 B -B 1 -B B .
X.12 17 . . 1 -1 -B B -1 2 -1 -5 -B B 1 B -B .
X.13 34 . . 2 2 . . 4 -1 -1 -6 . . . . . -1
X.14 34 . . 2 -2 . . 4 -1 -1 6 . . . . . 1
X.15 60 A *A -4 . . . . . . . . . . . . .
X.16 60 *A A -4 . . . . . . . . . . . . .
X.17 68 . . 4 . . . -4 -2 1 . . . . . . .
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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