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There is a growing demand for network support for group applications, in which messages from one or more sender(s) are delivered to a large number of receivers. Here, we propose a network architecture for supporting a fundamental type of group communication, conferencing. A conference refers to a group of members in a network who communicate with each other within the group. We consider adopting a class of multistage networks, such as a baseline, an omega, or an indirect binary cube network, composed of switch modules with fan-in and fan-out capability for a conference network which supports multiple disjoint conferences. The key issue in designing a conference network is to determine the multiplicity of routing conflicts, which is the maximum number of conflict parties competing a single interstage link when multiple disjoint conferences simultaneously present in the network. Our results show that, for a network of size n × n, the multiplicities of routing conflicts are small constants (between 2 and 4) for an omega network or an indirect binary cube network; while it can be as large as √n/q + 1 for a baseline network, where q is the minimum allowable conference size. Thus, our design for conference networks is based on an omega network or an indirect binary cube network. We also develop fast self-routing algorithms for setting up routing paths in the newly designed conference networks. As can be seen, such an n × n conference network has O(logn) routing time and communication delay and O(nlogn) hardware cost. The conference networks are superior to existing designs in terms of routing complexity, communication delay and hardware cost. The conference network proposed is rearrangeably nonblocking in general, and is strictly nonblocking under some conference service policy. It can be used in applications that require efficient or real-time group communication.