SAMS Subject Classification Number: 28

Let \(\overline{G}=P{.}G\) be a finite extension where \(P\) is a normal \(p\)-subgroup of \(\overline{G}\). Choose the smallest non-trivial characteristic subgroup \(K\) of \(P\) such that \(\frac{P}{K}=P_1\) is an abelian \(p\)-group then it follows that the factor group \(\overline{F}=\frac{\overline{G}}{K}\) has structure \(P_1{.}{G}\) and \(\overline{G}\) can be viewed as having the structure \(K{{}^\cdot}\overline{F}\). Since \(K\) is normal in \(\overline{G}\) the lifts of the ordinary characters Irr\((\overline{F})\) of \(\overline{F}\) are equivalent to ordinary irreducible characters \(\chi_i\) of \(\overline{G}\) such that \(K \leq \textrm{Ker}(\chi_i)\). Using the Fischer-Clifford matrices technique [1] we can obtain the ordinary character table of \(\overline{F}\). With the above discussion in mind, the Fischer-Clifford matrices \(M(g_i)\) of \(\overline{G}\), \(g_i\) a class representative of \(G\), can then be constructed from the Fischer-Clifford matrices \(\widetilde{M(g_i)}\) of \(\overline{F}\) by adding an appropriate number of columns and rows to \(\widetilde{M(g_i)}\). In a sense, the matrices \(\widetilde{M(g_i)}\) are "lifted" to the matrices \(M(g_i)\) in \(\overline{G}\). Having obtained the matrices \(M(g_i)\) and together with the ordinary or projective characters of the inertia factors \(H_i\) of the action of \(\overline{G}\) on Irr\((P)\), the ordinary irreducible character table of \(\overline{G}\) is constructed. The approach described above is powerful if \(P\) is non-abelian and \(P_1\) elementary abelian and a \(G\)-module over the finite field \(GF(p)\). In this talk, we will elaborate more about this approach and illustrate it with an appropriate example.