The Character Table of ${}^{2}E_{6}(2)$ Acting on the Cosets of $Fi_{22}$

Abstract

We consider the permutation action of $E \simeq {}^2E_6 (2)$ on the cosets of its maximal subgroup $F \simeq Fi_{22}$. We calculate the intersection matrices and character table of the centralizer algebra corresponding to this action. There are three reasons for the interest in this particular representation. Firstly, it is a sporadic multiplicity-free action of a simple group of exceptional Lie type. Secondly, $E$ and $F$ are $Y$-groups $Y_{333}$ and $Y_{332}$, respectively, factorized over their centers. We believe that the intersection matrices we have calculated might be useful for a computer-free identification of $Y_{333}$ with $2^3 \cdot {}^2E_6 (2)$. Thirdly, the permutation group considered is the one induced by the involution centralizer on the set of points fixed by an involution in the action of the Baby Monster $F_2$ on the cosets of the Fischer group $Fi_{23}$. The latter action has the largest rank (namely 23) among the primitive multiplicity-free actions of the sporadic simple groups and the calculation of its character table is an open problem.