Properties

Label 34T2
Order \(34\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{17}$

Learn more about

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $D_{17}$

Low degree siblings

17T2

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, 2, 2, 2, 2, 2 $ $17$ $2$ $( 1, 2)( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,28)(10,27)(11,26)(12,25) (13,24)(14,23)(15,21)(16,22)(17,19)(18,20)$
$ 17, 17 $ $2$ $17$ $( 1, 4, 6, 8, 9,12,14,15,18,19,22,24,26,27,29,31,34)( 2, 3, 5, 7,10,11,13,16, 17,20,21,23,25,28,30,32,33)$
$ 17, 17 $ $2$ $17$ $( 1, 6, 9,14,18,22,26,29,34, 4, 8,12,15,19,24,27,31)( 2, 5,10,13,17,21,25,30, 33, 3, 7,11,16,20,23,28,32)$
$ 17, 17 $ $2$ $17$ $( 1, 8,14,19,26,31, 4, 9,15,22,27,34, 6,12,18,24,29)( 2, 7,13,20,25,32, 3,10, 16,21,28,33, 5,11,17,23,30)$
$ 17, 17 $ $2$ $17$ $( 1, 9,18,26,34, 8,15,24,31, 6,14,22,29, 4,12,19,27)( 2,10,17,25,33, 7,16,23, 32, 5,13,21,30, 3,11,20,28)$
$ 17, 17 $ $2$ $17$ $( 1,12,22,31, 8,18,27, 4,14,24,34, 9,19,29, 6,15,26)( 2,11,21,32, 7,17,28, 3, 13,23,33,10,20,30, 5,16,25)$
$ 17, 17 $ $2$ $17$ $( 1,14,26, 4,15,27, 6,18,29, 8,19,31, 9,22,34,12,24)( 2,13,25, 3,16,28, 5,17, 30, 7,20,32,10,21,33,11,23)$
$ 17, 17 $ $2$ $17$ $( 1,15,29, 9,24, 4,18,31,12,26, 6,19,34,14,27, 8,22)( 2,16,30,10,23, 3,17,32, 11,25, 5,20,33,13,28, 7,21)$
$ 17, 17 $ $2$ $17$ $( 1,18,34,15,31,14,29,12,27, 9,26, 8,24, 6,22, 4,19)( 2,17,33,16,32,13,30,11, 28,10,25, 7,23, 5,21, 3,20)$

Group invariants

Order:  $34=2 \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [34, 1]
Character table:   
      2  1  1   .   .   .   .   .   .   .   .
     17  1  .   1   1   1   1   1   1   1   1

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

X.1      1  1   1   1   1   1   1   1   1   1
X.2      1 -1   1   1   1   1   1   1   1   1
X.3      2  .   A   E   D   B   F   G   H   C
X.4      2  .   B   C   F   A   D   H   G   E
X.5      2  .   C   A   H   E   G   D   F   B
X.6      2  .   D   G   C   F   E   A   B   H
X.7      2  .   E   B   G   C   H   F   D   A
X.8      2  .   F   H   E   D   C   B   A   G
X.9      2  .   G   F   A   H   B   E   C   D
X.10     2  .   H   D   B   G   A   C   E   F

A = E(17)^5+E(17)^12
B = E(17)^3+E(17)^14
C = E(17)^6+E(17)^11
D = E(17)^2+E(17)^15
E = E(17)^7+E(17)^10
F = E(17)^8+E(17)^9
G = E(17)^4+E(17)^13
H = E(17)+E(17)^16