The characters of the cubic surface group

The characters of the cubic surface group G are got by using its representation as a group of orthogonal projectivities in a four-dimensional space a over GF (3). In α , G permutes 27 pentagons transitively and 6 other characters, in addition to the permutation character of degree 27, are obtained by noting how the 5 vertices of each invariant pentagon are permuted. Other geometrical objects that afford transitive permutation representations are available, and provide material for similar investigations. The papers that have preceded this, wherein not only is the geometry described in detail but also many 5-rowed matrices of the orthogonal group are explicitly given, are laid under contribution. In particular, the table at the end of the most recent paper provides at sight not only permutation characters but monomial characters as well.