I encountered a number theory problem when doing my research:

1.I want to know whether or not there are infinitely many primes $p$ satistying $gcd(\frac{p-1}{6},6)=1$, such that $6$ is a cubic residue mod $p$, but $2$ and $3$ are not cubic residues mod $p$? If there are, can we give a expression of $p$?