Properties

Label 17T6
Order \(4080\)
n \(17\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,16)$

Related objects

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Group action invariants

Degree $n$ :  $17$
Transitive number $t$ :  $6$
Group :  $\PSL(2,16)$
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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

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 $ $1$ $1$ $()$
$ 5, 5, 5, 1, 1 $ $272$ $5$ $( 1,14, 4, 2, 5)( 3,17,11, 7,15)( 6, 9,13,10,12)$
$ 5, 5, 5, 1, 1 $ $272$ $5$ $( 1, 2,14, 5, 4)( 3, 7,17,15,11)( 6,10, 9,12,13)$
$ 3, 3, 3, 3, 3, 1, 1 $ $272$ $3$ $( 1,15,13)( 2,11, 6)( 3,10,14)( 4,17,12)( 5, 7, 9)$
$ 15, 1, 1 $ $272$ $15$ $( 1,10,17, 2, 9,15,14,12,11, 5,13, 3, 4, 6, 7)$
$ 15, 1, 1 $ $272$ $15$ $( 1, 3,12, 2, 7,13,14,17, 6, 5,15,10, 4,11, 9)$
$ 15, 1, 1 $ $272$ $15$ $( 1,12, 7,14, 6,15, 4, 9, 3, 2,13,17, 5,10,11)$
$ 15, 1, 1 $ $272$ $15$ $( 1,17, 9,14,11,13, 4, 7,10, 2,15,12, 5, 3, 6)$
$ 17 $ $240$ $17$ $( 1, 7, 4,14, 9, 2,11,10, 3, 8,15, 5, 6,16,12,17,13)$
$ 17 $ $240$ $17$ $( 1, 9, 3, 6,13,14,10, 5,17, 4,11,15,12, 7, 2, 8,16)$
$ 17 $ $240$ $17$ $( 1, 4, 9,11, 3,15, 6,12,13, 7,14, 2,10, 8, 5,16,17)$
$ 17 $ $240$ $17$ $( 1, 3,13,10,17,11,12, 2,16, 9, 6,14, 5, 4,15, 7, 8)$
$ 17 $ $240$ $17$ $( 1,11, 6, 7,10,16, 4, 3,12,14, 8,17, 9,15,13, 2, 5)$
$ 17 $ $240$ $17$ $( 1,10,12, 9, 5, 7, 3,17, 2, 6, 4, 8,13,11,16,14,15)$
$ 17 $ $240$ $17$ $( 1, 6,10, 4,12, 8, 9,13, 5,11, 7,16, 3,14,17,15, 2)$
$ 17 $ $240$ $17$ $( 1,12, 5, 3, 2, 4,13,16,15,10, 9, 7,17, 6, 8,11,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $255$ $2$ $( 1,10)( 2,12)( 3, 6)( 4,14)( 5, 7)( 9,16)(11,15)(13,17)$

Group invariants

Order:  $4080=2^{4} \cdot 3 \cdot 5 \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2  4   .   .   .   .   .   .   .   .  .  .  .   .   .   .   .  4
      3  1   .   .   .   .   .   .   .   .  1  1  1   1   1   1   1  .
      5  1   .   .   .   .   .   .   .   .  1  1  1   1   1   1   1  .
     17  1   1   1   1   1   1   1   1   1  .  .  .   .   .   .   .  .

        1a 17a 17b 17c 17d 17e 17f 17g 17h 5a 5b 3a 15a 15b 15c 15d 2a
     2P 1a 17c 17d 17b 17a 17g 17h 17f 17e 5b 5a 3a 15d 15c 15a 15b 1a
     3P 1a 17h 17g 17e 17f 17a 17b 17c 17d 5b 5a 1a  5b  5b  5a  5a 2a
     5P 1a 17g 17h 17f 17e 17b 17a 17d 17c 1a 1a 3a  3a  3a  3a  3a 2a
     7P 1a 17f 17e 17h 17g 17d 17c 17a 17b 5b 5a 3a 15c 15d 15b 15a 2a
    11P 1a 17e 17f 17g 17h 17c 17d 17b 17a 5a 5b 3a 15b 15a 15d 15c 2a
    13P 1a 17b 17a 17d 17c 17f 17e 17h 17g 5b 5a 3a 15d 15c 15a 15b 2a
    17P 1a  1a  1a  1a  1a  1a  1a  1a  1a 5b 5a 3a 15d 15c 15a 15b 2a

X.1      1   1   1   1   1   1   1   1   1  1  1  1   1   1   1   1  1
X.2     15   A   F   D   B   C   E   H   G  .  .  .   .   .   .   . -1
X.3     15   B   D   A   F   G   H   C   E  .  .  .   .   .   .   . -1
X.4     15   C   E   H   G   D   B   F   A  .  .  .   .   .   .   . -1
X.5     15   D   B   F   A   H   G   E   C  .  .  .   .   .   .   . -1
X.6     15   E   C   G   H   B   D   A   F  .  .  .   .   .   .   . -1
X.7     15   F   A   B   D   E   C   G   H  .  .  .   .   .   .   . -1
X.8     15   G   H   C   E   A   F   D   B  .  .  .   .   .   .   . -1
X.9     15   H   G   E   C   F   A   B   D  .  .  .   .   .   .   . -1
X.10    16  -1  -1  -1  -1  -1  -1  -1  -1  1  1  1   1   1   1   1  .
X.11    17   .   .   .   .   .   .   .   .  2  2 -1  -1  -1  -1  -1  1
X.12    17   .   .   .   .   .   .   .   .  I *I  2   I   I  *I  *I  1
X.13    17   .   .   .   .   .   .   .   . *I  I  2  *I  *I   I   I  1
X.14    17   .   .   .   .   .   .   .   .  I *I -1   J   K   M   L  1
X.15    17   .   .   .   .   .   .   .   .  I *I -1   K   J   L   M  1
X.16    17   .   .   .   .   .   .   .   . *I  I -1   L   M   J   K  1
X.17    17   .   .   .   .   .   .   .   . *I  I -1   M   L   K   J  1

A = -E(17)^3-E(17)^14
B = -E(17)^7-E(17)^10
C = -E(17)-E(17)^16
D = -E(17)^6-E(17)^11
E = -E(17)^4-E(17)^13
F = -E(17)^5-E(17)^12
G = -E(17)^8-E(17)^9
H = -E(17)^2-E(17)^15
I = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
J = E(15)^4+E(15)^11
K = E(15)+E(15)^14
L = E(15)^7+E(15)^8
M = E(15)^2+E(15)^13