In this paper, we study a kind of dual generalized inverses of dual matrices, which is called the dual group inverse. Some necessary and sufficient conditions for a dual matrix to have the dual group inverse are given. If one of these conditions is satisfied, then compact formulas and efficient methods for the computation of the dual group inverse are given. Moreover, the results of the dual group inverse are applied to solve systems of linear dual equations. The dual group-inverse solution of systems of linear dual equations is introduced. The dual analog of the real least-squares solution and minimal $P$-norm least-squares solution are obtained. Some numerical examples are provided to illustrate the results obtained.