Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.

$abcd=2^{10}\cdot 3^7\cdot 5^7$

$\mathrm{lcm}(a,b)=2^2\cdot 3^1\cdot 5^2$

$\mathrm{lcm}(a,c)=2^3\cdot 3^2\cdot 5^1$

$\mathrm{lcm}(a,d)=2^4\cdot 3^3\cdot 5^3$

$\mathrm{lcm}(b,c)=2^3\cdot 3^2\cdot 5^2$

$\mathrm{lcm}(b,d)=2^4\cdot 3^3\cdot 5^3$

$\mathrm{lcm}(c,d)=2^4\cdot 3^3\cdot 5^3$

What is $\gcd (a,b,c,d)$?