# Properties

 Label 17T4 Order $$136$$ n $$17$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $C_{17}:C_{8}$

# Related objects

## Group action invariants

 Degree $n$ : $17$ Transitive number $t$ : $4$ Group : $C_{17}:C_{8}$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7), (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:  $C_4$
8:  $C_8$

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

## Group invariants

 Order: $136=2^{3} \cdot 17$ Cyclic: No Abelian: No Solvable: Yes GAP id: [136, 12]
 Character table:  2 3 3 3 3 3 3 3 3 . . 17 1 . . . . . . . 1 1 1a 8a 4a 8b 8c 4b 8d 2a 17a 17b 2P 1a 4a 2a 4b 4b 2a 4a 1a 17a 17b 3P 1a 8b 4b 8a 8d 4a 8c 2a 17b 17a 5P 1a 8d 4a 8c 8b 4b 8a 2a 17b 17a 7P 1a 8c 4b 8d 8a 4a 8b 2a 17b 17a 11P 1a 8b 4b 8a 8d 4a 8c 2a 17b 17a 13P 1a 8d 4a 8c 8b 4b 8a 2a 17a 17b 17P 1a 8a 4a 8b 8c 4b 8d 2a 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 1 A -1 -A -A -1 A 1 1 1 X.4 1 -A -1 A A -1 -A 1 1 1 X.5 1 B -A -/B /B A -B -1 1 1 X.6 1 -/B A B -B -A /B -1 1 1 X.7 1 /B A -B B -A -/B -1 1 1 X.8 1 -B -A /B -/B A B -1 1 1 X.9 8 . . . . . . . C *C X.10 8 . . . . . . . *C C A = -E(4) = -Sqrt(-1) = -i B = -E(8) C = E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12+E(17)^14 = (-1-Sqrt(17))/2 = -1-b17