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We introduce a notion of c-group, which is a group up to congruence relationand consider the corresponding category. Extensions, actions and crossedmodules (c-crossed modules) are defined in this category and the semi-directproduct is constructed. We prove that each categorical group gives rise toc-groups and to a c-crossed module, which is a connected, special and strictc-crossed module in the sense defined by us. The results obtained here will beapplied in the proof of an equivalence of the categories of categorical groupsand connected, special and strict c-crossed modules.

Groups up to congruence relation and from categorical groups to c-crossed modules