Properties

Label 34T28
Degree $34$
Order $8160$
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)
$|\Aut(F/K)|$:  $2$
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)

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

Group invariants

Order:  $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table:   
      2  5  3  1  1   1   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 2a 5a 5b 10a 10b 17a 17b 17c 17d 2b 4a 3a 6a 15a 15b
     2P 1a 1a 5b 5a  5a  5b 17b 17a 17d 17c 1a 2b 3a 3a 15b 15a
     3P 1a 2a 5b 5a 10b 10a 17d 17c 17a 17b 2b 4a 1a 2a  5b  5a
     5P 1a 2a 1a 1a  2a  2a 17d 17c 17a 17b 2b 4a 3a 6a  3a  3a
     7P 1a 2a 5b 5a 10b 10a 17c 17d 17b 17a 2b 4a 3a 6a 15b 15a
    11P 1a 2a 5a 5b 10a 10b 17c 17d 17b 17a 2b 4a 3a 6a 15a 15b
    13P 1a 2a 5b 5a 10b 10a 17a 17b 17c 17d 2b 4a 3a 6a 15b 15a
    17P 1a 2a 5b 5a 10b 10a  1a  1a  1a  1a 2b 4a 3a 6a 15b 15a

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