We consider the description logic ALCFP(D_Σ) over the concrete domain D_Σ with the strict prefix order over finite strings. Using an automata-based approach, we show that the concept satisfiability problem w.r.t. general TBoxes for ALCFP(D_Σ) is ExpTime-complete for all finite alphabets Σ. As far as we know, this is the first complexity result for an expressive description logic with a nontrivial concrete domain on strings.