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Reaction with Lucas Reagent

What would be the product(s) if this alcohol reacts with Lucas reagent? Would Methyl shift happen ? I heard that in Groove's process, only primary and secondary alkyl chlorides are formed.

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A shift of the methyl group is likely, because this would yield a more stable tertiary carbocation. In concentrated hydrochloric acid, the double bond can also be protonated, producing a secondary carbocation which is captured by $\ce{Cl-}$. If both happens, the product would be 2,4-dichloro-4,5-dimethylhexane.

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