# Linear group L2(17)

Order = 2 448 = 24.32.17.
Mult = 2.
Out = 2.

Robert Wilson's ATLAS page for L2(17) is available here.

### L2(17): Length 69, 2-generator, 4-relator.

< x, y | x2 = y3 = (xy)17 = (xy)4(xy-1)2xy(xyxy-1)2(xy)2(xy-1)2 = 1 >

Remark: x and y are R.A.Wilson's standard generators for L2(17). The presentation is available in MAGMA code here (including the subgroups below too).

Some subgroups:

• M1 = < x, yxyxy-1xy-1xy > = 17:8 of order 136 and index 18.
• M2 = < x, yxyxyxyxy-1xyxy-1 > = S4 of order 24 and index 102.
• M3 = < x, yxyxy-1xyxyxyxy-1 > = S4 of order 24 and index 102.
• M4 = < x, xy > = D18 of order 18 and index 136.
• M5 = < x, xyxy-1xyxy > = D16 of order 16 and index 153.
• H1 = < x, xyxy > = D8 of order 8 and index 306.

Realisation: As permutations on 18 points in MAGMA format. As permutations of the projective line PG1(17) these are:
x: \eta -> -1 / \eta,
y: \eta -> \eta / (\eta - 1), and thus
xy: \eta -> \eta + 1.
In the MAGMA file we have used 17 for 0 and 18 for \infty.

### L2(17): Length 83, 2-generator, 4-relator.

< x, y | x2 = y3 = (xy)17 = ((xy)2xy-1(xy)7xy-1)2 = 1 >

Remark: Essentially a rewrite of the J.G.Sunday presentation. It is on the same (i.e, standard) generators as the previous presentation. The presentation is available in MAGMA code here (including some subgroups). For more information such as subgroups and realisations, see the version of this presentation above.

### L2(17) × 3: Length 35, 2-generator, 3-relator.

< x, y | x2 = y3 = (xy)4(xy-1)2xy(xyxy-1)2(xy)2(xy-1)2 = 1 >

Remark: x and y map on to standard generators for L2(17). The centre is generated by (xy)17. The presentation is available in MAGMA code here, including a subgroup H = < x, (yx)6y-1xy > = 17:8 over which a faithful permutation representation of degree 54 may be obtained. (The subgroup H2 turns out to be identical to H, but coset enumeration over this subgroup in the finitely presented group is harder.)

Realisation: As an intransitive group on 21 points (orbits 18 + 3) in MAGMA format.

Last updated 16th September, 1997
John N. Bray