Properties

 Label 17T8 Degree $17$ Order $16320$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSL(2,16):C_4$

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Group action invariants

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

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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 1$ $255$ $2$ $( 1,17)( 3,16)( 4,12)( 5, 7)( 6,15)( 8,13)( 9,14)(10,11)$ $4, 4, 4, 4, 1$ $1020$ $4$ $( 1, 7,17, 5)( 3,11,16,10)( 4,13,12, 8)( 6, 9,15,14)$ $8, 8, 1$ $2040$ $8$ $( 1,13, 5, 4,17, 8, 7,12)( 3, 9,10, 6,16,14,11,15)$ $8, 8, 1$ $2040$ $8$ $( 1,12, 7, 8,17, 4, 5,13)( 3,15,11,14,16, 6,10, 9)$ $17$ $960$ $17$ $( 1,16,13, 7, 3, 4,11, 9,15,17,14, 8,10, 5, 2, 6,12)$ $17$ $960$ $17$ $( 1,11,10,16, 9, 5,13,15, 2, 7,17, 6, 3,14,12, 4, 8)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $68$ $2$ $( 1,13)( 2, 6)( 4,12)( 5, 9)( 8,16)(10,14)$ $5, 5, 5, 1, 1$ $544$ $5$ $( 1, 8, 2,10,12)( 3,15,11,17, 7)( 4,13,16, 6,14)$ $10, 5, 2$ $1632$ $10$ $( 1,14, 8, 4, 2,13,10,16,12, 6)( 3,17,15, 7,11)( 5, 9)$ $3, 3, 3, 3, 3, 1, 1$ $272$ $3$ $( 1,17,12)( 2,16, 8)( 3, 4,10)( 5,13,15)( 7, 9,14)$ $15, 1, 1$ $1088$ $15$ $( 1, 3,16, 5, 9,17, 4, 8,13,14,12,10, 2,15, 7)$ $4, 4, 4, 2, 1, 1, 1$ $680$ $4$ $( 1, 5, 6,10)( 2, 4, 7, 8)( 9,11)(12,14,17,15)$ $4, 4, 4, 2, 1, 1, 1$ $680$ $4$ $( 1,10, 6, 5)( 2, 8, 7, 4)( 9,11)(12,15,17,14)$ $6, 6, 3, 1, 1$ $1360$ $6$ $( 1, 7,15, 6, 2,14)( 3,13,16)( 4,17, 5, 8,12,10)$ $12, 3, 2$ $1360$ $12$ $( 1,17, 7, 5,15, 8, 6,12, 2,10,14, 4)( 3,16,13)( 9,11)$ $12, 3, 2$ $1360$ $12$ $( 1,12, 7,10,15, 4, 6,17, 2, 5,14, 8)( 3,16,13)( 9,11)$

Group invariants

 Order: $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$ Cyclic: no Abelian: no Solvable: no GAP id: not available
 Character table:  2 6 1 2 . 4 1 . . 6 4 3 3 3 3 2 2 2 3 1 1 1 1 1 . . . . . . . 1 1 1 1 1 5 1 1 1 1 1 1 . . . . . . . . . . . 17 1 . . . . . 1 1 . . . . . . . . . 1a 5a 3a 15a 2a 10a 17a 17b 2b 4a 8a 8b 4b 4c 6a 12a 12b 2P 1a 5a 3a 15a 1a 5a 17a 17b 1a 2b 4a 4a 2a 2a 3a 6a 6a 3P 1a 5a 1a 5a 2a 10a 17b 17a 2b 4a 8b 8a 4c 4b 2a 4b 4c 5P 1a 1a 3a 3a 2a 2a 17b 17a 2b 4a 8a 8b 4b 4c 6a 12a 12b 7P 1a 5a 3a 15a 2a 10a 17b 17a 2b 4a 8b 8a 4c 4b 6a 12b 12a 11P 1a 5a 3a 15a 2a 10a 17b 17a 2b 4a 8b 8a 4c 4b 6a 12b 12a 13P 1a 5a 3a 15a 2a 10a 17a 17b 2b 4a 8a 8b 4b 4c 6a 12a 12b 17P 1a 5a 3a 15a 2a 10a 1a 1a 2b 4a 8a 8b 4b 4c 6a 12a 12b 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 1 1 -1 -1 1 1 1 -1 B -B -B B -1 B -B X.4 1 1 1 1 -1 -1 1 1 1 -1 -B B B -B -1 -B B X.5 16 1 1 1 4 -1 -1 -1 . . . . -2 -2 1 1 1 X.6 16 1 1 1 4 -1 -1 -1 . . . . 2 2 1 -1 -1 X.7 16 1 1 1 -4 1 -1 -1 . . . . C -C -1 B -B X.8 16 1 1 1 -4 1 -1 -1 . . . . -C C -1 -B B X.9 17 2 -1 -1 5 . . . 1 1 -1 -1 1 1 -1 1 1 X.10 17 2 -1 -1 5 . . . 1 1 1 1 -1 -1 -1 -1 -1 X.11 17 2 -1 -1 -5 . . . 1 -1 B -B B -B 1 -B B X.12 17 2 -1 -1 -5 . . . 1 -1 -B B -B B 1 B -B X.13 34 -1 4 -1 -6 -1 . . 2 2 . . . . . . . X.14 34 -1 4 -1 6 1 . . 2 -2 . . . . . . . X.15 60 . . . . . A *A -4 . . . . . . . . X.16 60 . . . . . *A A -4 . . . . . . . . X.17 68 -2 -4 1 . . . . 4 . . . . . . . . A = -E(17)-E(17)^2-E(17)^4-E(17)^8-E(17)^9-E(17)^13-E(17)^15-E(17)^16 = (1-Sqrt(17))/2 = -b17 B = -E(4) = -Sqrt(-1) = -i C = -2*E(4) = -2*Sqrt(-1) = -2i