Vectorization of Product of Three Matrices

Theorem
Let $R$ be a ring.

Let $A, B, C$ be matrices over $R$ such that the matrix product $ABC$ is defined.

Then $\operatorname{vec}(ABC) = (C^\intercal \otimes A) \cdot \operatorname{vec}(B)$ where:
 * $\operatorname{vec}$ denotes vectorization
 * $C^\intercal$ is the transpose of $C$
 * $\otimes$ denotes Kronecker product
 * $\cdot$ denotes matrix product

Also see

 * Vectorization of Product of Two Matrices