Group action invariants
| Degree $n$ : | $34$ | |
| Transitive number $t$ : | $33$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,20,16,22)(2,19,15,23)(3,18,14,24)(4,34,13,25)(5,33,12,26)(6,32,11,27)(7,31,10,28)(8,30,9,29)(17,21), (1,18,9,24)(2,23,8,19)(3,28,7,31)(4,33,6,26)(5,21)(10,29,17,30)(11,34,16,25)(12,22,15,20)(13,27,14,32) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 8: $D_{4}$ 16: $D_{8}$ 32: 32T51 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,33,31,29,27,25,23,21, 19,34,32,30,28,26,24,22,20)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,29,23,34,28,22,33,27, 21,32,26,20,31,25,19,30,24)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$ |
| $ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $32$ | $17$ | $(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,33,31,29,27,25,23,21, 19,34,32,30,28,26,24,22,20)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $289$ | $2$ | $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $578$ | $4$ | $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $2312$ | $4$ | $( 1,20,16,22)( 2,19,15,23)( 3,18,14,24)( 4,34,13,25)( 5,33,12,26)( 6,32,11,27) ( 7,31,10,28)( 8,30, 9,29)(17,21)$ |
| $ 8, 8, 8, 8, 1, 1 $ | $578$ | $8$ | $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$ |
| $ 8, 8, 8, 8, 1, 1 $ | $578$ | $8$ | $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$ |
| $ 16, 16, 1, 1 $ | $578$ | $16$ | $( 2, 6, 9, 7,14,15, 3,11,17,13,10,12, 5, 4,16, 8)(19,25,33,21,22,29,27,30,34, 28,20,32,31,24,26,23)$ |
| $ 16, 16, 1, 1 $ | $578$ | $16$ | $( 2,13, 9,12,14, 4, 3, 8,17, 6,10, 7, 5,15,16,11)(19,28,33,32,22,24,27,23,34, 25,20,21,31,29,26,30)$ |
| $ 16, 16, 1, 1 $ | $578$ | $16$ | $( 2,15,10, 8,14,13,16, 7,17, 4, 9,11, 5, 6, 3,12)(19,29,20,23,22,28,26,21,34, 24,33,30,31,25,27,32)$ |
| $ 16, 16, 1, 1 $ | $578$ | $16$ | $( 2, 4,10,11,14, 6,16,12,17,15, 9, 8, 5,13, 3, 7)(19,24,20,30,22,25,26,32,34, 29,33,23,31,28,27,21)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $2312$ | $4$ | $( 1,20,11,21)( 2,32,10,26)( 3,27, 9,31)( 4,22, 8,19)( 5,34, 7,24)( 6,29) (12,33,17,25)(13,28,16,30)(14,23,15,18)$ |
Group invariants
| Order: | $9248=2^{5} \cdot 17^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: |
2 5 . . . . . . . . . 5 4 2 4 4 4 4 4 4
17 2 2 2 2 2 2 2 2 2 2 . . . . . . . . .
1a 17a 17b 17c 17d 17e 17f 17g 17h 17i 2a 4a 4b 8a 8b 16a 16b 16c 16d
2P 1a 17c 17d 17a 17b 17e 17g 17f 17i 17h 1a 2a 2a 4a 4a 8b 8b 8a 8a
3P 1a 17b 17c 17d 17a 17e 17h 17i 17g 17f 2a 4a 4b 8b 8a 16d 16c 16a 16b
5P 1a 17b 17c 17d 17a 17e 17h 17i 17g 17f 2a 4a 4b 8b 8a 16c 16d 16b 16a
7P 1a 17d 17a 17b 17c 17e 17i 17h 17f 17g 2a 4a 4b 8a 8b 16b 16a 16d 16c
11P 1a 17d 17a 17b 17c 17e 17i 17h 17f 17g 2a 4a 4b 8b 8a 16c 16d 16b 16a
13P 1a 17a 17b 17c 17d 17e 17f 17g 17h 17i 2a 4a 4b 8b 8a 16d 16c 16a 16b
17P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 2a 4a 4b 8a 8b 16a 16b 16c 16d
X.1 1 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 -1 -1
X.3 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 1
X.4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1
X.5 2 2 2 2 2 2 2 2 2 2 2 2 . -2 -2 . . . .
X.6 2 2 2 2 2 2 2 2 2 2 -2 . . I -I J -J -K K
X.7 2 2 2 2 2 2 2 2 2 2 -2 . . I -I -J J K -K
X.8 2 2 2 2 2 2 2 2 2 2 -2 . . -I I K -K J -J
X.9 2 2 2 2 2 2 2 2 2 2 -2 . . -I I -K K -J J
X.10 2 2 2 2 2 2 2 2 2 2 2 -2 . . . I I -I -I
X.11 2 2 2 2 2 2 2 2 2 2 2 -2 . . . -I -I I I
X.12 32 -2 -2 -2 -2 15 -2 -2 -2 -2 . . . . . . . . .
X.13 32 A B D C -2 E H F G . . . . . . . . .
X.14 32 B D C A -2 F G H E . . . . . . . . .
X.15 32 C A B D -2 G F E H . . . . . . . . .
X.16 32 D C A B -2 H E G F . . . . . . . . .
X.17 32 E F H G -2 B C D A . . . . . . . . .
X.18 32 F H G E -2 D A C B . . . . . . . . .
X.19 32 G E F H -2 A D B C . . . . . . . . .
X.20 32 H G E F -2 C B A D . . . . . . . . .
2 2
17 .
4c
2P 2a
3P 4c
5P 4c
7P 4c
11P 4c
13P 4c
17P 4c
X.1 1
X.2 1
X.3 -1
X.4 -1
X.5 .
X.6 .
X.7 .
X.8 .
X.9 .
X.10 .
X.11 .
X.12 .
X.13 .
X.14 .
X.15 .
X.16 .
X.17 .
X.18 .
X.19 .
X.20 .
A = 2*E(17)^2+2*E(17)^3+2*E(17)^5+4*E(17)^6+4*E(17)^7+2*E(17)^8+2*E(17)^9+4*E(17)^10+4*E(17)^11+2*E(17)^12+2*E(17)^14+2*E(17)^15
B = 4*E(17)+2*E(17)^2+4*E(17)^4+2*E(17)^6+2*E(17)^7+2*E(17)^8+2*E(17)^9+2*E(17)^10+2*E(17)^11+4*E(17)^13+2*E(17)^15+4*E(17)^16
C = 2*E(17)+4*E(17)^2+2*E(17)^3+2*E(17)^4+2*E(17)^5+4*E(17)^8+4*E(17)^9+2*E(17)^12+2*E(17)^13+2*E(17)^14+4*E(17)^15+2*E(17)^16
D = 2*E(17)+4*E(17)^3+2*E(17)^4+4*E(17)^5+2*E(17)^6+2*E(17)^7+2*E(17)^10+2*E(17)^11+4*E(17)^12+2*E(17)^13+4*E(17)^14+2*E(17)^16
E = -E(17)-4*E(17)^2-2*E(17)^3-E(17)^4-2*E(17)^5-2*E(17)^6-2*E(17)^7-4*E(17)^8-4*E(17)^9-2*E(17)^10-2*E(17)^11-2*E(17)^12-E(17)^13-2*E(17)^14-4*E(17)^15-E(17)^16
F = -2*E(17)-2*E(17)^2-E(17)^3-2*E(17)^4-E(17)^5-4*E(17)^6-4*E(17)^7-2*E(17)^8-2*E(17)^9-4*E(17)^10-4*E(17)^11-E(17)^12-2*E(17)^13-E(17)^14-2*E(17)^15-2*E(17)^16
G = -2*E(17)-2*E(17)^2-4*E(17)^3-2*E(17)^4-4*E(17)^5-E(17)^6-E(17)^7-2*E(17)^8-2*E(17)^9-E(17)^10-E(17)^11-4*E(17)^12-2*E(17)^13-4*E(17)^14-2*E(17)^15-2*E(17)^16
H = -4*E(17)-E(17)^2-2*E(17)^3-4*E(17)^4-2*E(17)^5-2*E(17)^6-2*E(17)^7-E(17)^8-E(17)^9-2*E(17)^10-2*E(17)^11-2*E(17)^12-4*E(17)^13-2*E(17)^14-E(17)^15-4*E(17)^16
I = -E(8)+E(8)^3
= -Sqrt(2) = -r2
J = -E(16)+E(16)^7
K = -E(16)^3+E(16)^5
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