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We construct supplementary difference sets (SDS) with parameters$(72;36,30;30)$. These SDSs give periodic Golay pairs of length 72. No periodicGolay pair of length 72 was known previously. The smallest undecided order forperiodic Golay pairs is now 90. The periodic Golay pairs constructed here arethe first examples having length divisible by a prime congruent to 3 modulo 4.The main tool employed is a recently introduced compression method. We observethat Turyn's multiplication of Golay pairs can be also used to multiply a Golaypair and a periodic Golay pair.

Periodic Golay pairs of length 72