[vbiesnqs] Experimenting with larger groups

As computers have become more powerful and, in particular, have gotten more memory, has it become possible to easily do interesting experiments on certain finite groups which previously had been infeasible?

A group of order N can easily require N^2 memory for the "multiplication" table, and then N^3 or N^4 memory or time to run a computation over all elements or all multiplications.

Pedagogical experiments are interesting: groups beyond the cyclic group, especially the sporadic groups, often are difficult to understand.  Working with them through computer-aided experiments might help provide an intuitive understanding of them.  They lower the "barrier to entry" into advanced group theory, equivalently steepness of the learning curve.

Details of group presentation and group representation will probably be very important.

Inspired by Wikipedia article on the Monster group: "Robert A. Wilson has found explicitly (with the aid of a computer) two 196,882 by 196,882 matrices (with elements in the field of order 2) which together generate the monster group; this is one dimension lower than the 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices is possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes."  But in time, even "four and a half gigabytes" will become considered a trivial amount of memory and computation time.