\[\begin{aligned} ans&=\prod\limits_{i=1}^a\prod\limits_{j=1}^b\prod\limits_{k=1}^c(\frac{\operatorname{lcm}(i,j)}{\gcd(i,k)})^{f(i,j,k)}\\ &=\prod\limits_{i=1}^a\prod\limits_{j=1}^b\prod\limits_{k=1}^c(\frac{ij}{\gcd(i,j)\gcd(i,k)})^{f(i,j,k)}\\ \end{aligned} \]
\(\text{Let }P(a,b,c):=\prod\limits_{i=1}^a\prod\limits_{j=1}^b\prod\limits_{k=1}^ci^{f(i,j,k)},Q(a,b,c):=\prod\limits_{i=1}^a\prod\limits_{j=1}^b\prod\limits_{k=1}^c(\gcd(i,j))^{f(i,j,k)}\\ \text{Notice that }f(i,j,k)\text{ is rotating, so we have}\)
\[\begin{aligned} ans&=\prod\limits_{i=1}^a\prod\limits_{j=1}^b\prod\limits_{k=1}^c(\frac{ij}{\gcd(i,j)\gcd(i,k)})^{f(i,j,k)}\\ &=\frac{P(a,b,c)P(b,a,c)}{Q(a,b,c)Q(a,c,b)} \end{aligned} \]
\(\text{Thus what we need to calculate is }P(a,b,c)\text{ and }Q(a,b,c).\)
\(\text{For convenience, let }(f\times g)(n)\text{ denote }\prod\limits_{d|n}f(d)^{g(\frac nd)},(f\cdot g)(n)\text{ denote }f(n)^{g(n)}\text{ and }C(n)\text{ denote }\frac{n(n+1)}2.\)
\(\text{Sub.1:}f(i,j,k)=1\)
\[\begin{aligned} P(a,b,c)&=\prod\limits_{i=1}^a\prod\limits_{j=1}^b\prod\limits_{k=1}^ci\\ &=(a!)^{bc}\\ Q(a,b,c)&=\prod\limits_{i=1}^a\prod\limits_{j=1}^b\prod\limits_{k=1}^c\gcd(i,j)\\ &=(\prod\limits_{d=1}d^{\sum\limits_{t=1}\mu(t)\lfloor\frac a{dt}\rfloor\lfloor\frac b{dt}\rfloor})^c\\ &=\prod\limits_{d=1}(id\times \mu)(d)^{c\lfloor\frac ad\rfloor\lfloor\frac bd\rfloor} \end{aligned} \]
\(\text{Sub.2:}f(i,j,k)=ijk\)
\[\begin{aligned} P(a,b,c)&=\prod\limits_{i=1}^a\prod\limits_{j=1}^b\prod\limits_{k=1}^ci^{ijk}\\ &=(\prod\limits_{i=1}^ai^i)^{C(b)C(c)}\\ Q(a,b,c)&=\prod\limits_{i=1}^a\prod\limits_{j=1}^b\prod\limits_{k=1}^c\gcd(i,j)^{ijk}\\ &=(\prod\limits_{d=1}d^{d^2\sum\limits_{t=1}\mu(t)t^2C(\lfloor\frac a{dt}\rfloor)C(\lfloor\frac b{dt}\rfloor)})^{C(c)}\\ &=\prod\limits_{d=1}(id\times\mu\cdot id_2)(d)^{C(c)C(\lfloor\frac at\rfloor)C(\lfloor\frac bt\rfloor)} \end{aligned} \]