Properties

 Label 17T3 Degree $17$ Order $68$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{17}:C_{4}$

Related objects

Group action invariants

 Degree $n$: $17$ Transitive number $t$: $3$ Group: $C_{17}:C_{4}$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)

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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 4, 1$ $17$ $4$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)$ $4, 4, 4, 4, 1$ $17$ $4$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)$ $2, 2, 2, 2, 2, 2, 2, 2, 1$ $17$ $2$ $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$ $17$ $4$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$ $17$ $4$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)$ $17$ $4$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$ $17$ $4$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)$

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 4a 4b 2a 17a 17b 17c 17d 2P 1a 2a 2a 1a 17b 17a 17d 17c 3P 1a 4b 4a 2a 17c 17d 17b 17a 5P 1a 4a 4b 2a 17c 17d 17b 17a 7P 1a 4b 4a 2a 17d 17c 17a 17b 11P 1a 4b 4a 2a 17d 17c 17a 17b 13P 1a 4a 4b 2a 17a 17b 17c 17d 17P 1a 4a 4b 2a 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 A -A -1 1 1 1 1 X.4 1 -A A -1 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