ON A GROUP OF THE FORM \(2^{10} : (U_{5}(2) : 2)\)

Abstract

The full automorphism group U₅(2):2 of the special unitary group U₅(2) has a 10-dimensional absolutely irreducible module over GF(2).  Hence a split extension of the form \(\overline{G}\) = 2¹⁰:(U₅(2):2) does exist.  In this paper we first determine the conjugacy classes of \(\overline{G}\) using the coset analysis technique.  The structures of the inertia factor groups were determined.  These are the groups U₅(2):2, 2_¹⁺⁶:((3¹⁺²:8):2) and O₅(2):2.  We then determine the Fischer matrices and apply the Clifford-Fischer theory to compute the ordinary character table of \(\overline{G}\). The Fischer matrices \(\mathcal{F}\)_i of \(\overline{G}\) are all \(\mathbb{Z}\)-valued, with sizes range between 1 and 5.  The full character table of \(\overline{G}\), which is 109 × 109 \(\mathbb{C}\)-valued matrix is available in the PhD Thesis [1] of the first author, which could be accessed online.