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Abstract. We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction.

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If G is a finite primitive complex reflection group, all reflection subgroups of G and their inclusions are determined up to conjugacy. As a consequence, it is shown that if the rank of G is n and if G can be generated by n reflections, then for every set R of n reflections which generate G, every subset of R generates a parabolic subgroup of G. (4) 3 , J

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