The classes and Fischer–Clifford matrices of extensions $$p^{1+2n}{:}G$$ and their factor groups $$p^{2n}{:}G$$

Abstract Let $$\overline{G}=p^{1+2n}{:}G$$ be a finite split extension of an extra-special p -group $$P=p^{1+2n}$$ by a group G . Since the center Z ( P ) is characteristic in P and hence normal in $$\overline{G}$$ , we can construct the factor group $$\overline{F}=\frac{\overline{G}}{Z(P)}\cong p^{2n}{:}G$$ , where $$P_1=p^{2n}$$ is an elementary abelian p -group. In this paper, the Fischer–Clifford matrices M ( g ) of $$\overline{G}$$ are constructed from the corresponding Fischer–Clifford matrices $$\widehat{M(g)}$$ of $$\overline{F}$$ by a method we called the lifting of Fischer–Clifford matrices . As an example, the ordinary character table of a 7-local maximal subgroup $$7_{+}^{1+4}{:}(3\times 2 S_7)$$ of the Monster $$\mathbb {M}$$ is re-constructed using the lifting method.