# Properties

 Label 17T5 Degree $17$ Order $272$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $F_{17}$

# Related objects

## Group action invariants

 Degree $n$: $17$ Transitive number $t$: $5$ Group: $F_{17}$ Parity: $-1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (2,4,10,11,14,6,16,12,17,15,9,8,5,13,3,7), (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)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$
$16$:  $C_{16}$

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

## Group invariants

 Order: $272=2^{4} \cdot 17$ Cyclic: no Abelian: no Solvable: yes GAP id: [272, 50]
 Character table:  2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 . 17 1 . . . . . . . . . . . . . . . 1 1a 8a 16a 4a 16b 16c 16d 8b 8c 16e 16f 16g 4b 16h 8d 2a 17a 2P 1a 4a 8c 2a 8b 8a 8d 4b 4b 8d 8a 8b 2a 8c 4a 1a 17a 3P 1a 8b 16e 4b 16c 16g 16a 8a 8d 16h 16b 16f 4a 16d 8c 2a 17a 5P 1a 8d 16b 4a 16h 16d 16f 8c 8b 16c 16e 16a 4b 16g 8a 2a 17a 7P 1a 8c 16f 4b 16e 16h 16g 8d 8a 16b 16a 16d 4a 16c 8b 2a 17a 11P 1a 8b 16d 4b 16f 16b 16h 8a 8d 16a 16g 16c 4a 16e 8c 2a 17a 13P 1a 8d 16g 4a 16a 16e 16c 8c 8b 16f 16d 16h 4b 16b 8a 2a 17a 17P 1a 8a 16a 4a 16b 16c 16d 8b 8c 16e 16f 16g 4b 16h 8d 2a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 1 X.3 1 -1 A 1 A -A -A -1 -1 -A -A A 1 A -1 1 1 X.4 1 -1 -A 1 -A A A -1 -1 A A -A 1 -A -1 1 1 X.5 1 A B -1 -B /B -/B -A -A -/B /B -B -1 B A 1 1 X.6 1 A -B -1 B -/B /B -A -A /B -/B B -1 -B A 1 1 X.7 1 -A -/B -1 /B -B B A A B -B /B -1 -/B -A 1 1 X.8 1 -A /B -1 -/B B -B A A -B B -/B -1 /B -A 1 1 X.9 1 B C -A /D /C D -/B /B -D -/C -/D A -C -B -1 1 X.10 1 B -C -A -/D -/C -D -/B /B D /C /D A C -B -1 1 X.11 1 -/B D A -/C /D -C B -B C -/D /C -A -D /B -1 1 X.12 1 -/B -D A /C -/D C B -B -C /D -/C -A D /B -1 1 X.13 1 /B -/C A -D -C -/D -B B /D C D -A /C -/B -1 1 X.14 1 /B /C A D C /D -B B -/D -C -D -A -/C -/B -1 1 X.15 1 -B -/D -A C -D /C /B -/B -/C D -C A /D B -1 1 X.16 1 -B /D -A -C D -/C /B -/B /C -D C A -/D B -1 1 X.17 16 . . . . . . . . . . . . . . . -1 A = -E(4) = -Sqrt(-1) = -i B = -E(8) C = -E(16)^3 D = -E(16)