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Properties

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

Learn more about

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

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  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 3a 6a 5a 5b 15a 15b 2b 4a 17a 17b 17c 17d 10a 10b
     2P 1a 1a 3a 3a 5b 5a 15b 15a 1a 2b 17b 17a 17d 17c  5a  5b
     3P 1a 2a 1a 2a 5b 5a  5a  5b 2b 4a 17d 17c 17a 17b 10b 10a
     5P 1a 2a 3a 6a 1a 1a  3a  3a 2b 4a 17d 17c 17a 17b  2a  2a
     7P 1a 2a 3a 6a 5b 5a 15b 15a 2b 4a 17c 17d 17b 17a 10b 10a
    11P 1a 2a 3a 6a 5a 5b 15a 15b 2b 4a 17c 17d 17b 17a 10a 10b
    13P 1a 2a 3a 6a 5b 5a 15b 15a 2b 4a 17a 17b 17c 17d 10b 10a
    17P 1a 2a 3a 6a 5b 5a 15b 15a 2b 4a  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 -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 -1 -1  2  2  -1  -1  1  1   .   .   .   .   .   .
X.6     17 -5 -1  1  2  2  -1  -1  1 -1   .   .   .   .   .   .
X.7     17 -3  2  .  A *A  *A   A  1  1   .   .   .   .  *A   A
X.8     17 -3  2  . *A  A   A  *A  1  1   .   .   .   .   A  *A
X.9     17  3  2  .  A *A  *A   A  1 -1   .   .   .   . -*A  -A
X.10    17  3  2  . *A  A   A  *A  1 -1   .   .   .   .  -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  . -2  .  B *B -*A  -A  2  .   .   .   .   .   .   .
X.16    34  . -2  . *B  B  -A -*A  2  .   .   .   .   .   .   .

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