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.