  We present efficient quantum algorithms for the hidden subgroup problem (HSP)
on the semidirect product of cyclic groups $\Z_{p^r}\rtimes_{\phi}\Z_{p^2}$,
where $p$ is any odd prime number and $r$ is any integer such that $r>4$. We
also address the HSP in the group $\Z_{N}\rtimes_{\phi}\Z_{p^2}$, where $N$ is
an integer with a special prime factorization. These quantum algorithms are
exponentially faster than any classical algorithm for the same purpose.
