Properties

Label 17T7
Degree $17$
Order $8160$
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)
$|\Aut(F/K)|$:  $1$
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)

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

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

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  1  1  1   1   1 -4 -1   1   1  .  .
X.4     16  -1  -1  -1  -1  1  1  1   1   1  4  1  -1  -1  .  .
X.5     17   .   .   .   .  2  2 -1  -1  -1  5 -1   .   .  1  1
X.6     17   .   .   .   .  2  2 -1  -1  -1 -5  1   .   .  1 -1
X.7     17   .   .   .   .  E *E  2   E  *E -3  .   E  *E  1  1
X.8     17   .   .   .   . *E  E  2  *E   E -3  .  *E   E  1  1
X.9     17   .   .   .   .  E *E  2   E  *E  3  .  -E -*E  1 -1
X.10    17   .   .   .   . *E  E  2  *E   E  3  . -*E  -E  1 -1
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   .   .   .   .  F *F -2  -E -*E  .  .   .   .  2  .
X.16    34   .   .   .   . *F  F -2 -*E  -E  .  .   .   .  2  .

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