Computational group theory

In mathematics, computational group theory is the study of
groups by means of computers. It is concerned
with designing and analysing algorithms and
data structures to compute information about groups. The subject
has attracted interest because for many interesting groups
it is impractical
to perform calculations by hand.
Important algorithms in computational group theory include:

the Schreier–Sims algorithm for finding the order of a permutation group
the Todd–Coxeter algorithm and Knuth–Bendix algorithm for coset enumeration
the product-replacement algorithm for finding random elements of a group

Two important computer algebra systems used for group theory are
GAP and Magma. Historically, other systems such as CAS and Cayley were important.
Some achievements of the field include:

complete enumeration of all finite groups of order less than 2000
computation of representations for all the sporadic groups