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An approximate capacity result is demonstrated for a specific class of interference channels with common information (IC-CI). The class is the semideterministic interference channel of Telatar and Tse, and the outer bound herein generalizes their bound from the case where each transmitter sends only private information to the case where each transmitter sends both private and common information to its corresponding receiver. It is shown that our outer bound is within a quantifiable gap of the inner bound of Jiang-Xin-Garg, a generalization of the Han-Kobayashi region to the IC-CI. Moreover, this result reproduces both the Telatar-Tse result in the case of no common information and a constant-gap-to-capacity result for the Gaussian IC-CI.