Let G be a group and let S be a generating set of G . In this article,we introduce a metric d C on G with respect to S , called the cardinal metric.We then compare geometric structures of ( G , d C ) and ( G , d W ), where d W denotes the word metric. In particular, we prove that if S is finite, then ( G , d C ) and ( G , d W ) are not quasiisometric in the case when ( G , d W ) has infinite diameter and they are bi-Lipschitz equivalent otherwise. We also give an alternative description of cardinal metrics by using Cayley color graphs. It turns out that colorpermuting and color-preserving automorphisms of Cayley digraphs are isometries with respect to cardinal metrics.