Linear group L₂(32)

L₂(32): Length 67, 2-generator, 4-relator.

Remark: A presentation for L₂(32) on its unique class of (2, 3, 11)-generators (up to automorphisms). Generators are (2, 3, 11; 31). The presentation is available in MAGMA code here, where we have included the redundant relation ((xy)⁴xy^-1xyxy^-1xyxyxy^-1)² = 1 to aid coset enumeration. (We have included the subgroups below also.)

Some subgroups:

• M1 = < x, yxyxyxyxyxy^-1 > = 2⁵:31 of order 992 and index 33.
• M2 = < x, x^yxy > = D₆₆ of order 66 and index 496.
• M3 = < x, x^y > = D₆₂ of order 62 and index 528.

Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.

L₂(32): Length 83, 2-generator, 4-relator.

Remark: x and y are R.A.Wilson's standard generators for L₂(32). The generators are (2, 3, 31A/B/C/D/E; 11). Without (xyxy(xyxy^-1)⁴)² = 1, we get L₂(32) x 3. The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

• M1 = < x, yxyxy^-1xyxy^-1xy > = 2⁵:31 of order 992 and index 33.
• M2 = < x, x^yxy > = D₆₆ of order 66 and index 496.
• M3 = < x, x^yxyxy > = D₆₂ of order 62 and index 528.
• H1 = < x, x^y > = D₂₂ of order 22 and index 1 488.

Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.

L₂(32): Length 47, 2-generator, 3-relator.

Remark: A presentation of L₂(32) from (2, 3, 31F/G/H/I/J; 31)-generators such that xyxyxy^-1 has order 33. The presentation is available in MAGMA code here. (This applies to the subgroups below too.) I have added a redundant relation.

Some subgroups:

• M1 = < x, yxyxyxy^-1xyxy > = 2⁵:31 of order 992 and index 33.
• M2 = < x, x^yxyxy-1 > = D₆₆ of order 66 and index 496.
• M3 = < x, x^y > = D₆₂ of order 62 and index 528.
• H1 = < x, x^yxyxy > = D₂₂ of order 22 and index 1 488.

Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.

L₂(32): Length 39, 2-generator, 3-relator.

Remark: A presentation of L₂(32) from (2, 3, 31K/L/M/N/O; 31)-generators such that xyxyxy^-1 has order 11. The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

• M1 = < x, yxy^-1xyxyxyxy^-1 > = 2⁵:31 of order 992 and index 33.
• M2 = < x, x^yxyxy > = D₆₆ of order 66 and index 496.
• M3 = < x, x^y > = D₆₂ of order 62 and index 528.

Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.

L₂(32): Length 47, 2-generator, 3-relator.

Remark: A presentation of L₂(32) from (2, 3, 33A/B/C/D/E; 33)-generators such that xyxyxy^-1 has order 33. The presentation is available in MAGMA code here. (This applies to the subgroups below too.) I have added a redundant relation.

Some subgroups:

• M1 = < x, yxyxyxyxy^-1xy > = 2⁵:31 of order 992 and index 33.
• M2 = < x, x^y > = D₆₆ of order 66 and index 496.
• M3 = < x, x^yxy > = D₆₂ of order 62 and index 528.

Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.

L₂(32): Length 43, 2-generator, 3-relator.

Remark: A presentation of L₂(32) from (2, 3, 33F/G/H/I/J; 33)-generators such that xyxyxy^-1 has order 31. The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

• M1 = < x, yxyxyxy^-1xyxy^-1 > = 2⁵:31 of order 992 and index 33.
• M2 = < x, x^y > = D₆₆ of order 66 and index 496.
• M3 = < x, x^yxy > = D₆₂ of order 62 and index 528.

Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.

L₂(32): Length 58, 3-generator, 6-relator.

Remark: I'm afraid I don't know the source of this one (or at least of the presentation scheme of groups of type L₂(q) and PGL₂(q) whose presentations look similar to the one above). Traditionally, a, b and c are labelled as $\alpha$, $\beta$ and $\gamma$ in some order. The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

• M1 = < a, b > = 2⁵:31 of order 992 and index 33.
• M2 = < a, b^-3cb³ > = D₆₆ of order 66 and index 496.
• M3 = < b, c > = D₆₂ of order 62 and index 528.

Realisation:
a: \eta -> \eta + 1.
b: \eta -> w.\eta.
c: \eta -> -1 / \eta.

These linear fractional transformations are taken over the field F = GF(32) with primitive element w which satisfies w⁵ + w² + 1 = 0. (Strictly, they are taken over the projective line PG₁(32).)

The above linear fractional transformations have been converted into 2 x 2 matrices over GF(32) and are available in MAGMA format here.

L₂(32):5: Length 76, 4-generator, 10-relator.

Remark: The result of appending a field automorphism to one of the above presentations for L₂(32). The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

• M1 = < a, b, d > = 2⁵:(31:5) of order 4 960 and index 33.
• M2 = < a, (b⁶abc)^-1(ad)(b⁶abc) > = 33:10 of order 330 and index 496.
• M3 = < b, c, d > = 31:10 of order 310 and index 528.

Realisation:
a: \eta -> \eta + 1.
b: \eta -> w.\eta.
c: \eta -> -1 / \eta.
d: \eta -> \eta^2, a field automorphism (fixing $\infty$).

These linear fractional transformations are taken over the projective line PG₁(32) with underlying field F = GF(32) whose primitive element w satisfies w⁵ + w² + 1 = 0.

The above semi-linear fractional transformations have been converted into 10 x 10 matrices over GF(2) and are available in MAGMA format here.

L₂(32):5: Length 47, 2-generator, 4-relator.

Remark: A presentation of L₂(32):5 from a pair (x, y) of (2,5,5;33)-generators that satisfy o(x) = 2, o(y) = 5, o(xy) = 5, o([x, y²]) = 11, o([x, y²xy]) = 31. The presentation is available in MAGMA code here, and we have included the subgroups below also. I have included a number of redundant relations (mostly unactivated) to ease coset enumeration.

Some subgroups:

• M0 = < x, x^y, yxy^-1 > = L₂(32) of order 32 736 and index 5.
• M1 = < y, xyxy²x > = 2⁵:31:5 of order 4 960 and index 33.
• M2 = < x, y²xy²xy²xy > = 33:10 of order 330 and index 496.
• M3 = < x, yxy^-2xy^-2 > = 31:10 of order 310 and index 528.

Realisation: As 10 x 10 matrices over GF(2) in MAGMA format.

John N. Bray