In this study, we investigated the effect of social environment on circadian patterns in activity by group housing either six male or six female mice together in a cage, under regular light–dark cycles. Based on the interactions among the animals, the social dominance rank of individual mice was quantitatively established by calculating Elo ratings. Our results indicated that, during our experiment, the social dominance hierarchy was rapidly established, stable yet complex, often showing more than one dominant mouse and several subordinate mice. Moreover, we found that especially dominant male mice, but not female mice, displayed a significantly higher fraction of their activity during daytime. This resulted in reduced rhythm amplitude in dominant males. After division into separate cages, male mice showed an enhancement of their 24 h rhythm, due to lower daytime activity. Recordings of several physiological parameters showed no evidence for reduced health as a potential consequence of reduced rhythm amplitude. For female mice, transfer to individual housing did not affect their daily activity pattern. We conclude that 24 h rhythms under light–dark cycles are influenced by the social environment in males but not in females, and lead to a decrement in behavioural rhythm amplitude that is larger in dominant mice.