Properties

Label 17T8
Order \(16320\)
n \(17\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,16):C_4$

Related objects

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Group action invariants

Degree $n$ :  $17$
Transitive number $t$ :  $8$
Group :  $\PSL(2,16):C_4$
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), (1,12,7,5)(2,4,13,11)(3,8,10,14)(6,9), (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

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

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 $ $68$ $2$ $( 1, 5)( 4,14)( 6,16)( 7,12)( 8,11)( 9,17)$
$ 4, 4, 4, 2, 1, 1, 1 $ $680$ $4$ $( 1,14, 5, 4)( 3,10)( 6, 9,16,17)( 7, 8,12,11)$
$ 4, 4, 4, 2, 1, 1, 1 $ $680$ $4$ $( 1, 4, 5,14)( 3,10)( 6,17,16, 9)( 7,11,12, 8)$
$ 3, 3, 3, 3, 3, 1, 1 $ $272$ $3$ $( 1,16, 8)( 2, 3,10)( 4, 7, 9)( 5, 6,11)(12,17,14)$
$ 6, 6, 3, 1, 1 $ $1360$ $6$ $( 1,10,16, 2, 8, 3)( 4,11, 7, 5, 9, 6)(12,14,17)$
$ 12, 3, 2 $ $1360$ $12$ $( 1, 6,10, 4,16,11, 2, 7, 8, 5, 3, 9)(12,17,14)(13,15)$
$ 12, 3, 2 $ $1360$ $12$ $( 1, 7,10, 5,16, 9, 2, 6, 8, 4, 3,11)(12,17,14)(13,15)$
$ 5, 5, 5, 1, 1 $ $544$ $5$ $( 1, 5, 4, 2,14)( 3,12,16, 6, 7)( 8,11, 9,10,17)$
$ 15, 1, 1 $ $1088$ $15$ $( 1, 9,12, 5,10,16, 4,17, 6, 2, 8, 7,14,11, 3)$
$ 10, 5, 2 $ $1632$ $10$ $( 1,12, 2, 7, 5,16,14, 3, 4, 6)( 8,17,10, 9,11)(13,15)$
$ 17 $ $960$ $17$ $( 1, 7, 6, 3, 4,14, 2,16,11, 9, 5,15, 8,13,12,10,17)$
$ 17 $ $960$ $17$ $( 1, 2, 8, 7,16,13, 6,11,12, 3, 9,10, 4, 5,17,14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $255$ $2$ $( 1,17)( 2, 3)( 4, 6)( 5, 8)( 7, 9)(10,12)(11,15)(13,14)$
$ 4, 4, 4, 4, 1 $ $1020$ $4$ $( 1, 4,12,14)( 2,13, 7, 9)( 3,10,16, 5)( 6,15,17, 8)$
$ 8, 8, 1 $ $2040$ $8$ $( 1, 5, 4, 3,12,10,14,16)( 2,15,13,17, 7, 8, 9, 6)$
$ 8, 8, 1 $ $2040$ $8$ $( 1,16,14,10,12, 3, 4, 5)( 2, 6, 9, 8, 7,17,13,15)$

Group invariants

Order:  $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2  6  6  4  3  3  2  1   .  4   1   .   .  3  3  2   2   2
      3  1  .  .  .  .  1  1   1  1   .   .   .  1  1  1   1   1
      5  1  .  .  .  .  1  1   1  1   1   .   .  .  .  .   .   .
     17  1  .  .  .  .  .  .   .  .   .   1   1  .  .  .   .   .

        1a 2a 4a 8a 8b 3a 5a 15a 2b 10a 17a 17b 4b 4c 6a 12a 12b
     2P 1a 1a 2a 4a 4a 3a 5a 15a 1a  5a 17a 17b 2b 2b 3a  6a  6a
     3P 1a 2a 4a 8b 8a 1a 5a  5a 2b 10a 17b 17a 4c 4b 2b  4b  4c
     5P 1a 2a 4a 8a 8b 3a 1a  3a 2b  2b 17b 17a 4b 4c 6a 12a 12b
     7P 1a 2a 4a 8b 8a 3a 5a 15a 2b 10a 17b 17a 4c 4b 6a 12b 12a
    11P 1a 2a 4a 8b 8a 3a 5a 15a 2b 10a 17b 17a 4c 4b 6a 12b 12a
    13P 1a 2a 4a 8a 8b 3a 5a 15a 2b 10a 17a 17b 4b 4c 6a 12a 12b
    17P 1a 2a 4a 8a 8b 3a 5a 15a 2b 10a  1a  1a 4b 4c 6a 12a 12b

X.1      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  -1
X.3      1  1 -1  A -A  1  1   1 -1  -1   1   1 -A  A -1   A  -A
X.4      1  1 -1 -A  A  1  1   1 -1  -1   1   1  A -A -1  -A   A
X.5     16  .  .  .  .  1  1   1  4  -1  -1  -1 -2 -2  1   1   1
X.6     16  .  .  .  .  1  1   1  4  -1  -1  -1  2  2  1  -1  -1
X.7     16  .  .  .  .  1  1   1 -4   1  -1  -1  C -C -1   A  -A
X.8     16  .  .  .  .  1  1   1 -4   1  -1  -1 -C  C -1  -A   A
X.9     17  1  1 -1 -1 -1  2  -1  5   .   .   .  1  1 -1   1   1
X.10    17  1  1  1  1 -1  2  -1  5   .   .   . -1 -1 -1  -1  -1
X.11    17  1 -1  A -A -1  2  -1 -5   .   .   .  A -A  1  -A   A
X.12    17  1 -1 -A  A -1  2  -1 -5   .   .   . -A  A  1   A  -A
X.13    34  2  2  .  .  4 -1  -1 -6  -1   .   .  .  .  .   .   .
X.14    34  2 -2  .  .  4 -1  -1  6   1   .   .  .  .  .   .   .
X.15    60 -4  .  .  .  .  .   .  .   .   B  *B  .  .  .   .   .
X.16    60 -4  .  .  .  .  .   .  .   .  *B   B  .  .  .   .   .
X.17    68  4  .  .  . -4 -2   1  .   .   .   .  .  .  .   .   .

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(17)-E(17)^2-E(17)^4-E(17)^8-E(17)^9-E(17)^13-E(17)^15-E(17)^16
  = (1-Sqrt(17))/2 = -b17
C = -2*E(4)
  = -2*Sqrt(-1) = -2i