Is it possible to determine the group order by knowing the “public” and “private” key exponents in an RSA group?

Question

I have an RSA group with modulus $n = p \cdot q$, two safe primes $p=2p'+1$ and $q=2q'+1$ and the "public" and "private" key exponents $d$ and $e$. $\phi(n) = 4p'q'$ is the order of the RSA group. If I know $\phi(n)$ I can calculate $p$ and $q$. I ask myself what is if I know $e$ and $d$ (and $m$ and $n$) with $m^{d \cdot e\ \bmod\ \phi(n)}\ \bmod\ n$. Is it possible to calculate $\phi$ (and then $p$ and $q$)?