speeding up a special product of three arrays

I am struggling with R code optimization, a recurrent topic on this list.

I have three arrays, say A, B and C, all having the same number of columns.
I need to compute an array D whose generic element is

D[i, j, k] <- sum_n A[i, n]*B[j, n]*C[k, n]

Cycling over the three indices and subsetting the columns won't do. Is there
any way to implement this efficiently in R or should I resign to do this in
C?

Thanks,

Giuseppe

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