Group action invariants
Degree $n$: | $34$ | |
Transitive number $t$: | $44$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,30,28,16,11,21,24,32)(2,29,27,15,12,22,23,31)(3,34,19,17,26,6,8,9)(4,33,20,18,25,5,7,10)(13,14), (1,30,10,7,24,32,18,34)(2,29,9,8,23,31,17,33)(3,4)(5,21,15,27,11,25,19,13)(6,22,16,28,12,26,20,14) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $\PSL(2,16):C_4$
Low degree siblings
17T8Siblings are shown 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 17, 17 $ | $960$ | $17$ | $( 1,15, 3,29,14,18,19,33,28,26, 5,22,11,24,10,31, 8)( 2,16, 4,30,13,17,20,34, 27,25, 6,21,12,23, 9,32, 7)$ |
$ 17, 17 $ | $960$ | $17$ | $( 1,19,11,15,33,24, 3,28,10,29,26,31,14, 5, 8,18,22)( 2,20,12,16,34,23, 4,27, 9,30,25,32,13, 6, 7,17,21)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $68$ | $2$ | $( 3,10)( 4, 9)( 5,18)( 6,17)( 7,27)( 8,28)(13,20)(14,19)(15,29)(16,30)(25,32) (26,31)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $272$ | $3$ | $( 1,22,11)( 2,21,12)( 3,18,14)( 4,17,13)( 5,19,10)( 6,20, 9)( 7,32,16) ( 8,31,15)(25,30,27)(26,29,28)$ |
$ 6, 6, 6, 6, 3, 3, 1, 1, 1, 1 $ | $1360$ | $6$ | $( 1,11,22)( 2,12,21)( 3,19,18,10,14, 5)( 4,20,17, 9,13, 6)( 7,30,32,27,16,25) ( 8,29,31,28,15,26)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 $ | $680$ | $4$ | $( 1, 2)( 3, 7,10,27)( 4, 8, 9,28)( 5,25,18,32)( 6,26,17,31)(11,12) (13,15,20,29)(14,16,19,30)(21,22)(23,33)(24,34)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 $ | $680$ | $4$ | $( 1, 2)( 3,27,10, 7)( 4,28, 9, 8)( 5,32,18,25)( 6,31,17,26)(11,12) (13,29,20,15)(14,30,19,16)(21,22)(23,33)(24,34)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $255$ | $2$ | $( 3,10)( 4, 9)( 5,14)( 6,13)( 7,27)( 8,28)(11,22)(12,21)(15,26)(16,25)(17,20) (18,19)(23,34)(24,33)(29,31)(30,32)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $1020$ | $4$ | $( 3,24,10,33)( 4,23, 9,34)( 5,15,14,26)( 6,16,13,25)( 7,21,27,12)( 8,22,28,11) (17,32,20,30)(18,31,19,29)$ |
$ 8, 8, 8, 8, 2 $ | $2040$ | $8$ | $( 1, 2)( 3,30,24,17,10,32,33,20)( 4,29,23,18, 9,31,34,19)( 5, 7,15,21,14,27, 26,12)( 6, 8,16,22,13,28,25,11)$ |
$ 8, 8, 8, 8, 2 $ | $2040$ | $8$ | $( 1, 2)( 3,20,33,32,10,17,24,30)( 4,19,34,31, 9,18,23,29)( 5,12,26,27,14,21, 15, 7)( 6,11,25,28,13,22,16, 8)$ |
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $544$ | $5$ | $( 3,33,19,11,28)( 4,34,20,12,27)( 5, 8,31,29,10)( 6, 7,32,30, 9) (13,16,23,21,25)(14,15,24,22,26)$ |
$ 15, 15, 1, 1, 1, 1 $ | $1088$ | $15$ | $( 3, 5,24,33, 8,22,19,31,26,11,29,14,28,10,15)( 4, 6,23,34, 7,21,20,32,25,12, 30,13,27, 9,16)$ |
$ 12, 12, 6, 2, 2 $ | $1360$ | $12$ | $( 1,25, 3,17,29,13,28,21, 8,32,11,16)( 2,26, 4,18,30,14,27,22, 7,31,12,15) ( 5,23,10, 6,24, 9)(19,34)(20,33)$ |
$ 12, 12, 6, 2, 2 $ | $1360$ | $12$ | $( 1,21, 3,32,29,16,28,25, 8,17,11,13)( 2,22, 4,31,30,15,27,26, 7,18,12,14) ( 5,23,10, 6,24, 9)(19,34)(20,33)$ |
$ 10, 10, 5, 5, 2, 2 $ | $1632$ | $10$ | $( 1,19,10,14,18,28,26, 8,33,24)( 2,20, 9,13,17,27,25, 7,34,23)( 3,31,22,15,11) ( 4,32,21,16,12)( 5,29)( 6,30)$ |
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 4 2 2 3 3 2 2 6 4 3 3 1 1 . . . 3 1 1 1 1 1 1 1 1 . . . . 1 . 1 . . 5 1 1 1 . . . . . . . . . 1 1 1 . . 17 1 . . . . . . . . . . . . . . 1 1 1a 2a 3a 6a 4a 4b 12a 12b 2b 4c 8a 8b 5a 10a 15a 17a 17b 2P 1a 1a 3a 3a 2a 2a 6a 6a 1a 2b 4c 4c 5a 5a 15a 17a 17b 3P 1a 2a 1a 2a 4b 4a 4a 4b 2b 4c 8b 8a 5a 10a 5a 17b 17a 5P 1a 2a 3a 6a 4a 4b 12a 12b 2b 4c 8a 8b 1a 2a 3a 17b 17a 7P 1a 2a 3a 6a 4b 4a 12b 12a 2b 4c 8b 8a 5a 10a 15a 17b 17a 11P 1a 2a 3a 6a 4b 4a 12b 12a 2b 4c 8b 8a 5a 10a 15a 17b 17a 13P 1a 2a 3a 6a 4a 4b 12a 12b 2b 4c 8a 8b 5a 10a 15a 17a 17b 17P 1a 2a 3a 6a 4a 4b 12a 12b 2b 4c 8a 8b 5a 10a 15a 1a 1a 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 A -A -A A 1 -1 -A A 1 -1 1 1 1 X.4 1 -1 1 -1 -A A A -A 1 -1 A -A 1 -1 1 1 1 X.5 16 4 1 1 -2 -2 1 1 . . . . 1 -1 1 -1 -1 X.6 16 4 1 1 2 2 -1 -1 . . . . 1 -1 1 -1 -1 X.7 16 -4 1 -1 B -B A -A . . . . 1 1 1 -1 -1 X.8 16 -4 1 -1 -B B -A A . . . . 1 1 1 -1 -1 X.9 17 5 -1 -1 -1 -1 -1 -1 1 1 1 1 2 . -1 . . X.10 17 5 -1 -1 1 1 1 1 1 1 -1 -1 2 . -1 . . X.11 17 -5 -1 1 A -A -A A 1 -1 A -A 2 . -1 . . X.12 17 -5 -1 1 -A A A -A 1 -1 -A A 2 . -1 . . X.13 34 -6 4 . . . . . 2 2 . . -1 -1 -1 . . X.14 34 6 4 . . . . . 2 -2 . . -1 1 -1 . . X.15 60 . . . . . . . -4 . . . . . . C *C X.16 60 . . . . . . . -4 . . . . . . *C C X.17 68 . -4 . . . . . 4 . . . -2 . 1 . . A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i C = -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 |