The reflection group H_4 encodes the symmetries of the regular polytope called the 600-cell, which is a four-dimensional analogue of the icosahedron. The structure of H_4 has many interesting relations to a variety of topics of mathematics and physics. This book outlines the ideas behind the classification of finite reflection groups, giving a context for H_4 as the largest exceptional noncrystallographic reflection group. Different constructions of H_4 and its root system, which have thus far been published in pieces throughout the mathematics literature, are then described and compared. The book also examines a construction of the character table of H_4, which uses information about the group and root system structure. Finally, an algorithm which gives a systematic way of building character tables for any finite group is described. This book would be useful for anyone interested in finite reflection groups, especially those whose interests lie in the study of noncrystallographic groups.

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