Properties

 Label 17T7 Order $$8160$$ n $$17$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSL(2,16):C_2$

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) 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) $|\Aut(F/K)|$: $1$

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

Group invariants

 Order: $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$ Cyclic: No Abelian: No Solvable: No GAP id: Data not available
 Character table:  2 5 5 3 3 1 1 1 1 1 1 . . . . . . 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 4a 2b 5a 5b 10a 10b 3a 6a 17a 17b 17c 17d 15a 15b 2P 1a 1a 2a 1a 5b 5a 5a 5b 3a 3a 17b 17a 17d 17c 15b 15a 3P 1a 2a 4a 2b 5b 5a 10b 10a 1a 2b 17d 17c 17a 17b 5b 5a 5P 1a 2a 4a 2b 1a 1a 2b 2b 3a 6a 17d 17c 17a 17b 3a 3a 7P 1a 2a 4a 2b 5b 5a 10b 10a 3a 6a 17c 17d 17b 17a 15b 15a 11P 1a 2a 4a 2b 5a 5b 10a 10b 3a 6a 17c 17d 17b 17a 15a 15b 13P 1a 2a 4a 2b 5b 5a 10b 10a 3a 6a 17a 17b 17c 17d 15b 15a 17P 1a 2a 4a 2b 5b 5a 10b 10a 3a 6a 1a 1a 1a 1a 15b 15a 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 1 1 5 2 2 . . -1 -1 . . . . -1 -1 X.6 17 1 -1 -5 2 2 . . -1 1 . . . . -1 -1 X.7 17 1 -1 3 A *A -*A -A 2 . . . . . A *A X.8 17 1 -1 3 *A A -A -*A 2 . . . . . *A A X.9 17 1 1 -3 A *A *A A 2 . . . . . A *A X.10 17 1 1 -3 *A A A *A 2 . . . . . *A A X.11 30 -2 . . . . . . . . C D F E . . X.12 30 -2 . . . . . . . . D C E F . . X.13 30 -2 . . . . . . . . E F C D . . X.14 30 -2 . . . . . . . . F E D C . . X.15 34 2 . . B *B . . -2 . . . . . -A -*A X.16 34 2 . . *B B . . -2 . . . . . -*A -A 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