Properties

Label 34T4
Order \(68\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{17}.C_2$

Learn more about

Group action invariants

Degree $n$ :  $34$
Transitive number $t$ :  $4$
Group :  $D_{17}.C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,4,23)(2,16,3,24)(5,31,34,8)(6,32,33,7)(9,14,29,25)(10,13,30,26)(11,21,28,18)(12,22,27,17)(19,20), (1,14,25,4,16,27,5,18,30,7,19,32,10,21,34,12,24)(2,13,26,3,15,28,6,17,29,8,20,31,9,22,33,11,23)
$|\Aut(F/K)|$:  $2$

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: $C_{17}:C_{4}$

Low degree siblings

17T3

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

Group invariants

Order:  $68=2^{2} \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [68, 3]
Character table:   
     2  2  2  2  2   .   .   .   .
    17  1  .  .  .   1   1   1   1

       1a 2a 4a 4b 17a 17b 17c 17d
    2P 1a 1a 2a 2a 17b 17a 17d 17c
    3P 1a 2a 4b 4a 17c 17d 17b 17a
    5P 1a 2a 4a 4b 17c 17d 17b 17a
    7P 1a 2a 4b 4a 17d 17c 17a 17b
   11P 1a 2a 4b 4a 17d 17c 17a 17b
   13P 1a 2a 4a 4b 17a 17b 17c 17d
   17P 1a 2a 4a 4b  1a  1a  1a  1a

X.1     1  1  1  1   1   1   1   1
X.2     1  1 -1 -1   1   1   1   1
X.3     1 -1  A -A   1   1   1   1
X.4     1 -1 -A  A   1   1   1   1
X.5     4  .  .  .   B   E   C   D
X.6     4  .  .  .   C   D   E   B
X.7     4  .  .  .   D   C   B   E
X.8     4  .  .  .   E   B   D   C

A = -E(4)
  = -Sqrt(-1) = -i
B = E(17)^3+E(17)^5+E(17)^12+E(17)^14
C = E(17)^2+E(17)^8+E(17)^9+E(17)^15
D = E(17)+E(17)^4+E(17)^13+E(17)^16
E = E(17)^6+E(17)^7+E(17)^10+E(17)^11