Galois Group: 17T6

• Label: 17T6
• Order: \(4080\)
• n: \(17\)
• Cyclic: No
• Abelian: No
• Solvable: No
• Primitive: Yes
• $p$-group: No
• Group: $\PSL(2,16)$

Group action invariants

Degree $n$ :		$17$
Transitive number $t$ :		$6$
Group :		$\PSL(2,16)$
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)
$|\Aut(F/K)|$:		$1$

Low degree resolvents

None

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 $	$1$	$1$	$()$
$ 3, 3, 3, 3, 3, 1, 1 $	$272$	$3$	$( 1,11,13)( 2, 6,17)( 3,16, 8)( 4, 7,12)( 5, 9,15)$
$ 5, 5, 5, 1, 1 $	$272$	$5$	$( 1, 6, 9, 4, 8)( 2, 5,12,16,13)( 3,11,17,15, 7)$
$ 5, 5, 5, 1, 1 $	$272$	$5$	$( 1, 4, 6, 8, 9)( 2,16, 5,13,12)( 3,15,11, 7,17)$
$ 15, 1, 1 $	$272$	$15$	$( 1, 5, 3, 6,12,11, 9,16,17, 4,13,15, 8, 2, 7)$
$ 15, 1, 1 $	$272$	$15$	$( 1,15,16, 6, 7,13, 9, 3, 2, 4,11, 5, 8,17,12)$
$ 15, 1, 1 $	$272$	$15$	$( 1,16, 7, 9, 2,11, 8,12,15, 6,13, 3, 4, 5,17)$
$ 15, 1, 1 $	$272$	$15$	$( 1, 3,12, 9,17,13, 8, 7, 5, 6,11,16, 4,15, 2)$
$ 17 $	$240$	$17$	$( 1,17, 5, 3, 4, 8, 9,10,15, 2, 6,11,14,13,12,16, 7)$
$ 17 $	$240$	$17$	$( 1, 4,15,14, 7, 3,10,11,16, 5, 9, 6,12,17, 8, 2,13)$
$ 17 $	$240$	$17$	$( 1, 5, 4, 9,15, 6,14,12, 7,17, 3, 8,10, 2,11,13,16)$
$ 17 $	$240$	$17$	$( 1,15, 7,10,16, 9,12, 8,13, 4,14, 3,11, 5, 6,17, 2)$
$ 17 $	$240$	$17$	$( 1, 9,14,17,10,13, 5,15,12, 3, 2,16, 4, 6, 7, 8,11)$
$ 17 $	$240$	$17$	$( 1,10,12, 4,11,17,15,16, 8,14, 5, 2, 7, 9,13, 3, 6)$
$ 17 $	$240$	$17$	$( 1,14,10, 5,12, 2, 4, 7,11, 9,17,13,15, 3,16, 6, 8)$
$ 17 $	$240$	$17$	$( 1,12,11,15, 8, 5, 7,13, 6,10, 4,17,16,14, 2, 9, 3)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$255$	$2$	$( 1, 2)( 3, 9)( 4, 8)( 5,10)( 6, 7)(11,16)(12,14)(15,17)$

Group invariants

Order:		$4080=2^{4} \cdot 3 \cdot 5 \cdot 17$
Cyclic:		No
Abelian:		No
Solvable:		No
GAP id:		Data not available

Character table:

2 4 . . . . . . . . . . . . . . . 4 3 1 1 1 1 1 1 1 1 . . . . . . . . . 5 1 1 1 1 1 1 1 1 . . . . . . . . . 17 1 . . . . . . . 1 1 1 1 1 1 1 1 . 1a 3a 5a 5b 15a 15b 15c 15d 17a 17b 17c 17d 17e 17f 17g 17h 2a 2P 1a 3a 5b 5a 15d 15c 15a 15b 17c 17d 17b 17a 17g 17h 17f 17e 1a 3P 1a 1a 5b 5a 5a 5a 5b 5b 17h 17g 17e 17f 17a 17b 17c 17d 2a 5P 1a 3a 1a 1a 3a 3a 3a 3a 17g 17h 17f 17e 17b 17a 17d 17c 2a 7P 1a 3a 5b 5a 15c 15d 15b 15a 17f 17e 17h 17g 17d 17c 17a 17b 2a 11P 1a 3a 5a 5b 15b 15a 15d 15c 17e 17f 17g 17h 17c 17d 17b 17a 2a 13P 1a 3a 5b 5a 15d 15c 15a 15b 17b 17a 17d 17c 17f 17e 17h 17g 2a 17P 1a 3a 5b 5a 15d 15c 15a 15b 1a 1a 1a 1a 1a 1a 1a 1a 2a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 15 . . . . . . . F K I G H J M L -1 X.3 15 . . . . . . . G I F K L M H J -1 X.4 15 . . . . . . . H J M L I G K F -1 X.5 15 . . . . . . . I G K F M L J H -1 X.6 15 . . . . . . . J H L M G I F K -1 X.7 15 . . . . . . . K F G I J H L M -1 X.8 15 . . . . . . . L M H J F K I G -1 X.9 15 . . . . . . . M L J H K F G I -1 X.10 16 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 . X.11 17 -1 2 2 -1 -1 -1 -1 . . . . . . . . 1 X.12 17 2 A *A *A *A A A . . . . . . . . 1 X.13 17 2 *A A A A *A *A . . . . . . . . 1 X.14 17 -1 A *A B C D E . . . . . . . . 1 X.15 17 -1 A *A C B E D . . . . . . . . 1 X.16 17 -1 *A A D E C B . . . . . . . . 1 X.17 17 -1 *A A E D B C . . . . . . . . 1 A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = E(15)^7+E(15)^8 C = E(15)^2+E(15)^13 D = E(15)^4+E(15)^11 E = E(15)+E(15)^14 F = -E(17)^3-E(17)^14 G = -E(17)^7-E(17)^10 H = -E(17)-E(17)^16 I = -E(17)^6-E(17)^11 J = -E(17)^4-E(17)^13 K = -E(17)^5-E(17)^12 L = -E(17)^8-E(17)^9 M = -E(17)^2-E(17)^15