Properties

Label 34T33
Order \(9248\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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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 TypeSizeOrderRepresentative
$ 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