# Properties

 Label 17T7 Degree $17$ Order $8160$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSL(2,16):C_2$

This is the transitive group of least degree $d$ for which no irreducible polynomial in $\Q[x]$ of degree $d$ with this Galois group is known.

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $68$ $2$ $( 1,13)( 3,12)( 5,17)( 6, 7)( 8,15)(11,14)$ $5, 5, 5, 1, 1$ $272$ $5$ $( 1, 5,15, 7,12)( 2,10,16, 4, 9)( 3,13,17, 8, 6)$ $5, 5, 5, 1, 1$ $272$ $5$ $( 1, 7, 5,12,15)( 2, 4,10, 9,16)( 3, 8,13, 6,17)$ $10, 5, 2$ $816$ $10$ $( 1, 6, 5, 3,15,13, 7,17,12, 8)( 2, 4,10, 9,16)(11,14)$ $10, 5, 2$ $816$ $10$ $( 1,17,15, 6,12,13, 5, 8, 7, 3)( 2,10,16, 4, 9)(11,14)$ $17$ $480$ $17$ $( 1,15,17,13, 3, 9,10,14, 6,16, 7, 5, 2, 4,12, 8,11)$ $17$ $480$ $17$ $( 1,17, 3,10, 6, 7, 2,12,11,15,13, 9,14,16, 5, 4, 8)$ $17$ $480$ $17$ $( 1,10, 2,15,14, 4,17, 6,12,13,16, 8, 3, 7,11, 9, 5)$ $17$ $480$ $17$ $( 1, 2,14,17,12,16, 3,11, 5,10,15, 4, 6,13, 8, 7, 9)$ $3, 3, 3, 3, 3, 1, 1$ $272$ $3$ $( 1,17,16)( 4, 7, 8)( 5, 9, 6)(10,13,15)(11,14,12)$ $15, 1, 1$ $544$ $15$ $( 1,13, 9, 7,14,17,15, 6, 8,12,16,10, 5, 4,11)$ $15, 1, 1$ $544$ $15$ $( 1, 6,11,15, 4,17, 5,14,10, 7,16, 9,12,13, 8)$ $2, 2, 2, 2, 2, 2, 2, 2, 1$ $255$ $2$ $( 1,11)( 2, 4)( 3,12)( 5,17)( 6, 8)( 7,15)( 9,16)(13,14)$ $4, 4, 4, 4, 1$ $1020$ $4$ $( 1, 8,11, 6)( 2,17, 4, 5)( 3,16,12, 9)( 7,14,15,13)$ $6, 6, 3, 1, 1$ $1360$ $6$ $( 1, 6,13,16, 5,15)( 2, 4,12, 3, 7,14)( 8,10,11)$

## Group invariants

 Order: $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$ Cyclic: no Abelian: no Solvable: no Label: not available
 Character table:  2 5 3 1 1 1 1 . . 1 1 . . . . 5 3 3 1 1 1 1 1 1 1 1 . . . . . . . . 5 1 1 1 . 1 1 1 1 1 1 . . . . . . 17 1 . . . . . . . . . 1 1 1 1 . . 1a 2a 3a 6a 5a 5b 15a 15b 10a 10b 17a 17b 17c 17d 2b 4a 2P 1a 1a 3a 3a 5b 5a 15b 15a 5b 5a 17b 17a 17d 17c 1a 2b 3P 1a 2a 1a 2a 5b 5a 5a 5b 10b 10a 17d 17c 17a 17b 2b 4a 5P 1a 2a 3a 6a 1a 1a 3a 3a 2a 2a 17d 17c 17a 17b 2b 4a 7P 1a 2a 3a 6a 5b 5a 15b 15a 10b 10a 17c 17d 17b 17a 2b 4a 11P 1a 2a 3a 6a 5a 5b 15a 15b 10a 10b 17c 17d 17b 17a 2b 4a 13P 1a 2a 3a 6a 5b 5a 15b 15a 10b 10a 17a 17b 17c 17d 2b 4a 17P 1a 2a 3a 6a 5b 5a 15b 15a 10b 10a 1a 1a 1a 1a 2b 4a X.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 X.3 16 -4 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 . . X.4 16 4 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 . . X.5 17 5 -1 -1 2 2 -1 -1 . . . . . . 1 1 X.6 17 -5 -1 1 2 2 -1 -1 . . . . . . 1 -1 X.7 17 -3 2 . A *A *A A A *A . . . . 1 1 X.8 17 -3 2 . *A A A *A *A A . . . . 1 1 X.9 17 3 2 . A *A *A A -A -*A . . . . 1 -1 X.10 17 3 2 . *A A A *A -*A -A . . . . 1 -1 X.11 30 . . . . . . . . . C D F E -2 . X.12 30 . . . . . . . . . D C E F -2 . X.13 30 . . . . . . . . . E F C D -2 . X.14 30 . . . . . . . . . F E D C -2 . X.15 34 . -2 . B *B -*A -A . . . . . . 2 . X.16 34 . -2 . *B B -A -*A . . . . . . 2 . A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 C = -E(17)^6-E(17)^7-E(17)^10-E(17)^11 D = -E(17)^3-E(17)^5-E(17)^12-E(17)^14 E = -E(17)-E(17)^4-E(17)^13-E(17)^16 F = -E(17)^2-E(17)^8-E(17)^9-E(17)^15