The perpendicular bisector of the line segment  with end points$\left(2,3,2\right)$ and $\left(-4,1,4\right)$ passes through the point  $\left(-3,6,1\right)$and has equation of the form $\frac{x+3}{a}=\frac{y-6}{b}=\frac{z-1}{c}$where $a,b,c$ are relatively prime integers with $a>0$ , then the value of $abc-\left(a+b+c\right)$ is

The midpoint of the line segment joining $\left(2,3,2\right)$,$\left(-4,1,4\right)$is$\left(\frac{2-4}{2},\frac{3+1}{2},\frac{2+4}{2}\right)=\left(-1,2,3\right)$  

Given that the perpendicular bisector is passing through the point $\left(-3,6,1\right)$ also

Hence the equation of the perpendicular bisector is  $\frac{x+3}{-1+3}=\frac{y-6}{2-6}=\frac{z-1}{3-1}$

Rewrite the above equation as  $\frac{x+3}{-1+3}=\frac{y-6}{2-6}=\frac{z-1}{3-1}$

Comparing this with the given line$a=1,b=-2,c=1$

Consider the expression $abc-\left(a+b+c\right)=-2-0=\boxed{-2}$