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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This is the transitive group of least degree $d$ for which no irreducible polynomial in $\Q[x]$ of degree $d$ with this Galois group is known.

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)
$\card{\Aut(F/K)}$:  $1$
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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $68$ $2$ $( 1,13)( 3,12)( 5,17)( 6, 7)( 8,15)(11,14)$
$ 5, 5, 5, 1, 1 $ $272$ $5$ $( 1, 5,15, 7,12)( 2,10,16, 4, 9)( 3,13,17, 8, 6)$
$ 5, 5, 5, 1, 1 $ $272$ $5$ $( 1, 7, 5,12,15)( 2, 4,10, 9,16)( 3, 8,13, 6,17)$
$ 10, 5, 2 $ $816$ $10$ $( 1, 6, 5, 3,15,13, 7,17,12, 8)( 2, 4,10, 9,16)(11,14)$
$ 10, 5, 2 $ $816$ $10$ $( 1,17,15, 6,12,13, 5, 8, 7, 3)( 2,10,16, 4, 9)(11,14)$
$ 17 $ $480$ $17$ $( 1,15,17,13, 3, 9,10,14, 6,16, 7, 5, 2, 4,12, 8,11)$
$ 17 $ $480$ $17$ $( 1,17, 3,10, 6, 7, 2,12,11,15,13, 9,14,16, 5, 4, 8)$
$ 17 $ $480$ $17$ $( 1,10, 2,15,14, 4,17, 6,12,13,16, 8, 3, 7,11, 9, 5)$
$ 17 $ $480$ $17$ $( 1, 2,14,17,12,16, 3,11, 5,10,15, 4, 6,13, 8, 7, 9)$
$ 3, 3, 3, 3, 3, 1, 1 $ $272$ $3$ $( 1,17,16)( 4, 7, 8)( 5, 9, 6)(10,13,15)(11,14,12)$
$ 15, 1, 1 $ $544$ $15$ $( 1,13, 9, 7,14,17,15, 6, 8,12,16,10, 5, 4,11)$
$ 15, 1, 1 $ $544$ $15$ $( 1, 6,11,15, 4,17, 5,14,10, 7,16, 9,12,13, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $255$ $2$ $( 1,11)( 2, 4)( 3,12)( 5,17)( 6, 8)( 7,15)( 9,16)(13,14)$
$ 4, 4, 4, 4, 1 $ $1020$ $4$ $( 1, 8,11, 6)( 2,17, 4, 5)( 3,16,12, 9)( 7,14,15,13)$
$ 6, 6, 3, 1, 1 $ $1360$ $6$ $( 1, 6,13,16, 5,15)( 2, 4,12, 3, 7,14)( 8,10,11)$

Group invariants

Order:  $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$
Cyclic:  no
Abelian:  no
Solvable:  no
Label:  not available
Character table:   
      2  5  3  1  1  1  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 2a 3a 6a 5a 5b 15a 15b 10a 10b 17a 17b 17c 17d 2b 4a
     2P 1a 1a 3a 3a 5b 5a 15b 15a  5b  5a 17b 17a 17d 17c 1a 2b
     3P 1a 2a 1a 2a 5b 5a  5a  5b 10b 10a 17d 17c 17a 17b 2b 4a
     5P 1a 2a 3a 6a 1a 1a  3a  3a  2a  2a 17d 17c 17a 17b 2b 4a
     7P 1a 2a 3a 6a 5b 5a 15b 15a 10b 10a 17c 17d 17b 17a 2b 4a
    11P 1a 2a 3a 6a 5a 5b 15a 15b 10a 10b 17c 17d 17b 17a 2b 4a
    13P 1a 2a 3a 6a 5b 5a 15b 15a 10b 10a 17a 17b 17c 17d 2b 4a
    17P 1a 2a 3a 6a 5b 5a 15b 15a 10b 10a  1a  1a  1a  1a 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 -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   A  *A   .   .   .   .  1  1
X.8     17 -3  2  . *A  A   A  *A  *A   A   .   .   .   .  1  1
X.9     17  3  2  .  A *A  *A   A  -A -*A   .   .   .   .  1 -1
X.10    17  3  2  . *A  A   A  *A -*A  -A   .   .   .   .  1 -1
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  . -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