On groups with an irredundant 7-cover

A cover for a group is a collection of proper subgroups whose union is the whole group. A cover is irredundant if no proper sub-collection is also a cover and is called maximal if all its members are maximal subgroups. For an integer n>2n>2, a cover with nn members is called an nn-cover. In this paper we determine groups with a maximal irredundant 7-cover   with core-free intersection. The intersection of an irredundant nn-cover   is known to have index bounded by a function of nn, though in general the precise bound is not known. Here we prove that the exact bound is 81 when nn is 7.