In this paper, we investigate a novel finite-time median-related group consensus problem, where the finial consensus value can be identified as a desired function of the median of initial states instead of the much studied average value. The underlying communication topology is modelled by a weighted dynamical directed network. A distributed control protocol is firstly introduced to ensure that the agents can reach a median-related consensus in finite time in a collaboration network, meaning that all edge-weights of the communication network are non-negative. We then generalise the results to cooperation-competition networks, where the communication network is divided into predetermined collaboration subnetworks allowing possibly negative weights. Effective group control protocols are designed to guarantee the median-related group consensus in finite time. Finally, numerical simulations are presented to illustrate the availability of our theoretical results.