Properties

Label 34T28
Order \(8160\)
n \(34\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $34$
Transitive number $t$ :  $28$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,29,2,6,30)(3,23,33,15,20,9)(4,24,34,16,19,10)(7,21,25,13,32,17)(8,22,26,14,31,18)(11,12)(27,28), (1,25,18,32,7,16,29,14,10,6,3,33,22,20,24,28,12)(2,26,17,31,8,15,30,13,9,5,4,34,21,19,23,27,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $\PSL(2,16):C_2$

Low degree siblings

17T7

Siblings are shown 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $255$ $2$ $( 1,12)( 2,11)( 3,33)( 4,34)( 5,19)( 6,20)( 7,18)( 8,17)( 9,30)(10,29)(13,21) (14,22)(23,26)(24,25)(27,31)(28,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1020$ $4$ $( 1, 4,12,34)( 2, 3,11,33)( 5,14,19,22)( 6,13,20,21)( 7,30,18, 9)( 8,29,17,10) (15,16)(23,32,26,28)(24,31,25,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $68$ $2$ $( 1, 2)( 3,34)( 4,33)( 5,32)( 6,31)( 7, 8)( 9,29)(10,30)(11,12)(13,24)(14,23) (15,16)(17,18)(19,28)(20,27)(21,25)(22,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $272$ $3$ $( 1,16,18)( 2,15,17)( 3,24,29)( 4,23,30)( 5,21,19)( 6,22,20)( 9,34,13) (10,33,14)(25,28,32)(26,27,31)$
$ 6, 6, 6, 6, 6, 2, 2 $ $1360$ $6$ $( 1,17,16, 2,18,15)( 3, 9,24,34,29,13)( 4,10,23,33,30,14)( 5,28,21,32,19,25) ( 6,27,22,31,20,26)( 7, 8)(11,12)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $272$ $5$ $( 1,32,33, 3, 6)( 2,31,34, 4, 5)( 9,30,19,17,27)(10,29,20,18,28) (13,23,21,15,26)(14,24,22,16,25)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $272$ $5$ $( 1, 3,32, 6,33)( 2, 4,31, 5,34)( 9,17,30,27,19)(10,18,29,28,20) (13,15,23,26,21)(14,16,24,25,22)$
$ 10, 10, 10, 2, 2 $ $816$ $10$ $( 1,21, 3,13,32,15, 6,23,33,26)( 2,22, 4,14,31,16, 5,24,34,25)( 7,11)( 8,12) ( 9,28,17,20,30,10,27,18,19,29)$
$ 10, 10, 10, 2, 2 $ $816$ $10$ $( 1,23,32,21,33,15, 3,26, 6,13)( 2,24,31,22,34,16, 4,25, 5,14)( 7,11)( 8,12) ( 9,18,30,28,19,10,17,29,27,20)$
$ 15, 15, 1, 1, 1, 1 $ $544$ $15$ $( 1,29,25, 6,10,16, 3,28,22,33,18,24,32,20,14)( 2,30,26, 5, 9,15, 4,27,21,34, 17,23,31,19,13)$
$ 15, 15, 1, 1, 1, 1 $ $544$ $15$ $( 1,28,14, 3,20,16,32,10,24, 6,18,25,33,29,22)( 2,27,13, 4,19,15,31, 9,23, 5, 17,26,34,30,21)$
$ 17, 17 $ $480$ $17$ $( 1,25, 7,20,32,22,28,12, 6,16, 3,24,10,18,29,33,14)( 2,26, 8,19,31,21,27,11, 5,15, 4,23, 9,17,30,34,13)$
$ 17, 17 $ $480$ $17$ $( 1, 7,32,28, 6, 3,10,29,14,25,20,22,12,16,24,18,33)( 2, 8,31,27, 5, 4, 9,30, 13,26,19,21,11,15,23,17,34)$
$ 17, 17 $ $480$ $17$ $( 1,28,10,25,12,18, 7, 6,29,20,16,33,32, 3,14,22,24)( 2,27, 9,26,11,17, 8, 5, 30,19,15,34,31, 4,13,21,23)$
$ 17, 17 $ $480$ $17$ $( 1,10,12, 7,29,16,32,14,24,28,25,18, 6,20,33, 3,22)( 2, 9,11, 8,30,15,31,13, 23,27,26,17, 5,19,34, 4,21)$

Group invariants

Order:  $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2  5  1  1  1  3  1  5  3   .   .   .   .   .   .   1   1
      3  1  1  1  1  1  1  .  .   1   1   .   .   .   .   .   .
      5  1  1  1  1  1  .  .  .   1   1   .   .   .   .   1   1
     17  1  .  .  .  .  .  .  .   .   .   1   1   1   1   .   .

        1a 5a 5b 3a 2a 6a 2b 4a 15a 15b 17a 17b 17c 17d 10a 10b
     2P 1a 5b 5a 3a 1a 3a 1a 2b 15b 15a 17b 17a 17d 17c  5a  5b
     3P 1a 5b 5a 1a 2a 2a 2b 4a  5a  5b 17d 17c 17a 17b 10b 10a
     5P 1a 1a 1a 3a 2a 6a 2b 4a  3a  3a 17d 17c 17a 17b  2a  2a
     7P 1a 5b 5a 3a 2a 6a 2b 4a 15b 15a 17c 17d 17b 17a 10b 10a
    11P 1a 5a 5b 3a 2a 6a 2b 4a 15a 15b 17c 17d 17b 17a 10a 10b
    13P 1a 5b 5a 3a 2a 6a 2b 4a 15b 15a 17a 17b 17c 17d 10b 10a
    17P 1a 5b 5a 3a 2a 6a 2b 4a 15b 15a  1a  1a  1a  1a 10b 10a

X.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
X.3     16  1  1  1 -4 -1  .  .   1   1  -1  -1  -1  -1   1   1
X.4     16  1  1  1  4  1  .  .   1   1  -1  -1  -1  -1  -1  -1
X.5     17  2  2 -1  5 -1  1  1  -1  -1   .   .   .   .   .   .
X.6     17  2  2 -1 -5  1  1 -1  -1  -1   .   .   .   .   .   .
X.7     17  A *A  2 -3  .  1  1  *A   A   .   .   .   .  *A   A
X.8     17 *A  A  2 -3  .  1  1   A  *A   .   .   .   .   A  *A
X.9     17  A *A  2  3  .  1 -1  *A   A   .   .   .   . -*A  -A
X.10    17 *A  A  2  3  .  1 -1   A  *A   .   .   .   .  -A -*A
X.11    30  .  .  .  .  . -2  .   .   .   C   D   F   E   .   .
X.12    30  .  .  .  .  . -2  .   .   .   D   C   E   F   .   .
X.13    30  .  .  .  .  . -2  .   .   .   E   F   C   D   .   .
X.14    30  .  .  .  .  . -2  .   .   .   F   E   D   C   .   .
X.15    34  B *B -2  .  .  2  . -*A  -A   .   .   .   .   .   .
X.16    34 *B  B -2  .  .  2  .  -A -*A   .   .   .   .   .   .

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
B = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5
C = -E(17)^6-E(17)^7-E(17)^10-E(17)^11
D = -E(17)^3-E(17)^5-E(17)^12-E(17)^14
E = -E(17)-E(17)^4-E(17)^13-E(17)^16
F = -E(17)^2-E(17)^8-E(17)^9-E(17)^15