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Groups and FieldsSuperchallengeThe Sylow Theorems

The Sylow Theorems · Superchallenge problem

A group has order 30030=2⋅3⋅5⋅7⋅11⋅1330030=2\cdot3\cdot5\cdot7\cdot11\cdot13, the product of the first six primes. If its Sylow 1313-subgroup is NOT normal, what is the LARGEST value Sylow III still allows for n13n_{13}?