Properties

Label 17T3
Order \(68\)
n \(17\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{17}:C_{4}$

Related objects

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Group action invariants

Degree $n$ :  $17$
Transitive number $t$ :  $3$
Group :  $C_{17}:C_{4}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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 TypeSizeOrderRepresentative
$ 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