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Firstly we present new generating sets for the condensed algebra, which are often small enough to be computationally tractable. We also introduce a method which enables us to decrease the size of such a generating set by verifying the redundancy of generators without prior condensation. Secondly we develop a new criterion intheory and practice for proving that a subset of the condensed algebra is a set of generators. In the final chapter of this thesis we combine our new condensation methods with our knowledge of generating systems for the condensed algebra to compute the 2- and 3-modular character tables of the sporadic simple Fischer group Fi22 as part of the modular atlas project.","abstract_html":"This thesis contributes to the algorithmic representation theory of sporadic simple groups. We augment the condensation method by allowing to condense with an arbitrary primitive idempotent of a linear character of the condensation subgroup. 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