# Properties

 Label 34T4 Degree $34$ Order $68$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{17}.C_2$

## Group action invariants

 Degree $n$: $34$ Transitive number $t$: $4$ Group: $D_{17}.C_2$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ 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)

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

## 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