There is a nice survey of the subject area available in pdf: Survey: Computational Group Theory, which while somewhat dated, gives a nice introduction to the field and provides some historical insights.
Here's a very nice Introduction to Computational Group Theory. It's a brief but fascinating survey by Ákos Seress, published in the AMS Notices (1997: 06).
See also, of course, Wikipedia: Computational Group Theory. It's not a very lengthy entry, but there are nice references provided, and links that expand a bit more on what is discussed. Wikipedia mentions two computer algebra systems: GAP and Magma, which each have links to learn more. They are incredibly useful, powerful, and time-saving systems that enriches the study of group theory. GAP is freely available from its website, and also as part of the SAGE system, which is free to download as well, but can also be used online.
Derek F. Holt $\dagger$, Bettina Eick, Bettina, Eamonn A. O'Brien, "Handbook of computational group theory", Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005.
Charles C. Sims, "Computation with Finitely-presented Groups," Encyclopedia of
Mathematics and its Applications, vol 48, Cambridge University Press,
Ákos Seress, "Permutation group algorithms", Cambridge Tracts in
Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003.
$\dagger$ Note that author Derek F. Holt is a regular contributer on Math.SE, and has provided a nice link in a comment below!