Properties

Label 34T44
Order \(16320\)
n \(34\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $34$
Transitive number $t$ :  $44$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,30,28,16,11,21,24,32)(2,29,27,15,12,22,23,31)(3,34,19,17,26,6,8,9)(4,33,20,18,25,5,7,10)(13,14), (1,30,10,7,24,32,18,34)(2,29,9,8,23,31,17,33)(3,4)(5,21,15,27,11,25,19,13)(6,22,16,28,12,26,20,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $\PSL(2,16):C_4$

Low degree siblings

17T8

Siblings are shown 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $68$ $2$ $( 1,31)( 2,32)( 5,33)( 6,34)( 9,20)(10,19)(13,21)(14,22)(15,26)(16,25)(17,27) (18,28)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $272$ $3$ $( 1,10,14)( 2, 9,13)( 3, 8,11)( 4, 7,12)( 5,15,18)( 6,16,17)(19,22,31) (20,21,32)(25,27,34)(26,28,33)$
$ 6, 6, 6, 6, 3, 3, 1, 1, 1, 1 $ $1360$ $6$ $( 1,22,10,31,14,19)( 2,21, 9,32,13,20)( 3,11, 8)( 4,12, 7)( 5,28,15,33,18,26) ( 6,27,16,34,17,25)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 $ $680$ $4$ $( 1,34,31, 6)( 2,33,32, 5)( 3, 4)( 7, 8)( 9,26,20,15)(10,25,19,16)(11,12) (13,28,21,18)(14,27,22,17)(23,29)(24,30)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 $ $680$ $4$ $( 1, 6,31,34)( 2, 5,32,33)( 3, 4)( 7, 8)( 9,15,20,26)(10,16,19,25)(11,12) (13,18,21,28)(14,17,22,27)(23,29)(24,30)$
$ 12, 12, 6, 2, 2 $ $1360$ $12$ $( 1,27,19, 6,14,25,31,17,10,34,22,16)( 2,28,20, 5,13,26,32,18, 9,33,21,15) ( 3,12, 8, 4,11, 7)(23,29)(24,30)$
$ 12, 12, 6, 2, 2 $ $1360$ $12$ $( 1,17,19,34,14,16,31,27,10, 6,22,25)( 2,18,20,33,13,15,32,28, 9, 5,21,26) ( 3,12, 8, 4,11, 7)(23,29)(24,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $255$ $2$ $( 1,22)( 2,21)( 3,11)( 4,12)( 5,28)( 6,27)( 9,20)(10,19)(13,32)(14,31)(15,26) (16,25)(17,34)(18,33)(23,30)(24,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $1020$ $4$ $( 1,28,22, 5)( 2,27,21, 6)( 3,15,11,26)( 4,16,12,25)( 9,30,20,23)(10,29,19,24) (13,17,32,34)(14,18,31,33)$
$ 8, 8, 8, 8, 2 $ $2040$ $8$ $( 1,23,28, 9,22,30, 5,20)( 2,24,27,10,21,29, 6,19)( 3,17,15,32,11,34,26,13) ( 4,18,16,31,12,33,25,14)( 7, 8)$
$ 8, 8, 8, 8, 2 $ $2040$ $8$ $( 1,20, 5,30,22, 9,28,23)( 2,19, 6,29,21,10,27,24)( 3,13,26,34,11,32,15,17) ( 4,14,25,33,12,31,16,18)( 7, 8)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $544$ $5$ $( 1,29,26,31,15)( 2,30,25,32,16)( 3,10,18,11, 5)( 4, 9,17,12, 6) ( 7,27,20,23,13)( 8,28,19,24,14)$
$ 10, 10, 5, 5, 2, 2 $ $1632$ $10$ $( 1,31,29,15,26)( 2,32,30,16,25)( 3,24,10,14,18, 8,11,28, 5,19) ( 4,23, 9,13,17, 7,12,27, 6,20)(21,34)(22,33)$
$ 15, 15, 1, 1, 1, 1 $ $1088$ $15$ $( 1,19,14, 8,11,18,24,33,15,26, 3,31, 5,28,10)( 2,20,13, 7,12,17,23,34,16,25, 4,32, 6,27, 9)$
$ 17, 17 $ $960$ $17$ $( 1,15,29,28,10,14,33, 3,31,22,19,26, 8,18, 5,11,24)( 2,16,30,27, 9,13,34, 4, 32,21,20,25, 7,17, 6,12,23)$
$ 17, 17 $ $960$ $17$ $( 1,33, 8,15, 3,18,29,31, 5,28,22,11,10,19,24,14,26)( 2,34, 7,16, 4,17,30,32, 6,27,21,12, 9,20,23,13,25)$

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

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i
C = -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