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:
Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.
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:
Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.
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:
Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.
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:
Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.
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:
Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.
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:
Realisation: As 2 x 2 matrices over GF(32) in MAGMA format.
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:
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.
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:
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.
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:
Realisation: As 10 x 10 matrices over GF(2) in MAGMA format.