Properties

Label 17T7
Order \(8160\)
n \(17\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,16):C_2$

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Group action invariants

Degree $n$ :  $17$
Transitive number $t$ :  $7$
Group :  $\PSL(2,16):C_2$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,3)(4,9)(5,7)(6,8)(10,14)(11,13)(12,15)(16,17), (1,16)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15), (1,6,13,5,4,2,15,10,14,12,3,9,7,11,8), (1,7)(2,13)(3,10)(4,11)(5,12)(8,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $68$ $2$ $( 4, 6)( 5, 7)( 8, 9)(10,11)(12,15)(13,14)$
$ 3, 3, 3, 3, 3, 1, 1 $ $272$ $3$ $( 1, 2,16)( 4, 9,14)( 5,10,12)( 6, 8,13)( 7,11,15)$
$ 6, 6, 3, 1, 1 $ $1360$ $6$ $( 1,16, 2)( 4,13, 9, 6,14, 8)( 5,15,10, 7,12,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $255$ $2$ $( 1,16)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)$
$ 4, 4, 4, 4, 1 $ $1020$ $4$ $( 1, 9,16, 8)( 2,10, 3,11)( 4,14, 5,15)( 6,12, 7,13)$
$ 5, 5, 5, 1, 1 $ $272$ $5$ $( 1, 5, 7,15,12)( 2, 6,11, 8,16)( 3, 9,14, 4,17)$
$ 5, 5, 5, 1, 1 $ $272$ $5$ $( 1,15, 5,12, 7)( 2, 8, 6,16,11)( 3, 4, 9,17,14)$
$ 10, 5, 2 $ $816$ $10$ $( 1,16,12, 8,15,11, 7, 6, 5, 2)( 3,14,17, 9, 4)(10,13)$
$ 10, 5, 2 $ $816$ $10$ $( 1, 6,15,16, 5,11,12, 2, 7, 8)( 3,17, 4,14, 9)(10,13)$
$ 17 $ $480$ $17$ $( 1,17, 7,12,16,15, 8, 2, 3,11,13, 5, 4,14, 9, 6,10)$
$ 17 $ $480$ $17$ $( 1, 7,16, 8, 3,13, 4, 9,10,17,12,15, 2,11, 5,14, 6)$
$ 17 $ $480$ $17$ $( 1, 8, 4,17, 2,14, 7, 3, 9,12,11, 6,16,13,10,15, 5)$
$ 17 $ $480$ $17$ $( 1, 4, 2, 7, 9,11,16,10, 5, 8,17,14, 3,12, 6,13,15)$
$ 15, 1, 1 $ $544$ $15$ $( 1, 7, 6,14, 5, 2,11, 8, 4,10,16,15,13, 9,12)$
$ 15, 1, 1 $ $544$ $15$ $( 1, 8,12,11, 9, 2,13, 5,15,14,16, 6,10, 7, 4)$

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