ON AN INERTIA FACTOR GROUP OF 2\(^{8}\):\(O_8^{+}\)(2)

Abstract

ON AN INERTIA FACTOR GROUP OF 2\(^{8}\):\(O_n ^{+}\)(2)
 
The group \(\bar{G}\) = 2⁶:A₈ is an inertia factor group of 2\(^{8}\):\(O_8^{+}\)(2).  As an inertia factor group, our group \(\bar{G}\) plays an essential role in the construction of the character table of 2\(^{8}\):\(O_n ^{+}\)(2).  In this paper we look at two ways of constructing \(\bar{G}\).  In the first method, we use combinatorics and the natural action of A₈ on W \(\cong\) 2₆.  In the second method, we use a computational method to construct \(\bar{G}\) = N:A₈ inside \(O_8^{+}\)(2).  We show that A₈ acts irreducibly on both W and N and we prove that the two groups are indeed isomorphic.