On a Maximal Parabolic Subgroup of $$O^{+}_{8}(2)$$ O 8 + ( 2 )

Abstract

The simple orthogonal group $$O^{+}_{8}(2)$$ has a maximal parabolic subgroup of the form $$\bar{G}=N{:}G$$ , where $$N=2^{6}$$ and $$G\cong A_{8}$$ . Using Atlas, we can see that $$O^{+}_{8}(2)$$ has three maximal subgroups of type $$2^6{:}A_{8}$$ . These three maximal subgroups are conjugate in the full automorphism group of $$O^{+}_{8}(2)$$ , namely $$O^{+}_{8}(2){:}S_{3}$$ . Thus, we have the modules $$M_{1}=2^{6}$$ , $$M_{2}=2^{6}$$ , and $$M_{3}=2^{6}$$ on which $$A_{8}$$ acts irreducibly (therefore, these three modules are isomorphic as $$O^{+}_{8}(2)$$ -module). Without loss of generality, we can consider our maximal parabolic subgroup to be $$\bar{G}=M_{1}{:}A_{8}$$ . The Fischer matrices for each class representative of G are computed which together with character tables of inertia factor groups of G lead to the full character table of $$\bar{G}$$ . The complete fusion of $$\bar{G}$$ into $$O^{+}_{8}(2)$$ has been determined using the technique of set intersections of characters.