Higher Indicators for the Doubles of some Totally Orthogonal Groups

We investigate the indicators for certain groups of the form $\BZ_k\rtimes D_l$andtheirdoubles,where$D_l$isthedihedralgroupoforder$2l$. We subsequently obtain an infinite family of totally orthogonal, completely real groups which are generated by involutions, and whose doubles admit modules with second indicator of -1. This provides us with answers to several questions concerning the doubles of totally orthogonal finite groups.