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

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

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