Linear group L₃(5)

L₃(5): Length 59, 2-generator, 5-relator.

Remark: x and y are R.A.Wilson's standard generators for L₃(5). The presentation is available in MAGMA code here. (This applies to the subgroups below too.) I have included a redundant relation in the presentation to aid coset enumeration.

Some subgroups:

• M1 = < y, xy²xyx^-1 > = 5²:GL₂(5) of order 12 000 and index 31.
• M2 = < y, x^-1yxy²x > = 5²:GL₂(5) of order 12 000 and index 31.
• M3 = < x, y^-1xyxyxyxy^-1 > = S₅ of order 120 and index 3 100.
• M4 = < x, yx^-1y²x^-1y²x^-1y > = 4²:S₃ of order 96 and index 3 875.
• M5 = < x, yx^-1yxy^-1xy^-1 > = 31:3 of order 93 and index 4 000.
• H1 = < x, yx^-1y²x^-1y² > = 5²:GL₂(5) of order 12 000 and index 31.
• H2 = < x, y²x^-1y²x^-1y > = 5²:GL₂(5) of order 12 000 and index 31.
• H3 = < x, yxy²xy^-2xy^-1xy > = S₅ of order 120 and index 3 100.

Realisation: As 3 × 3 matrices over GF(5) in MAGMA format.

L₃(5):2: Length ??, 2-generator, ?-relator.