Question

Product of cycles.
Is (2497)(64259)(16)(376)=(1963257)(4)=(1963257) ?

Answer 1

(376) fixes 1, (16) sends 1 to 6, (64259) sends 6 to 4, and (2497) sends 4 to 9. So in total 1 goes to 9. Thus far (19...)

(376) fixes 9, (16) fixes 9, (64259) sends 9 to 6, and (2497) fixes 6. So in total 9 goes to 6. Thus far (196...)

(376) sends 6 to 3, (16) fixes 3, (64259) fixes 3, and (2497) fixes 3. So in total 6 goes to 3. Thus far (1963...)

(376) sends 3 to 7, (16) fixes 7, (64259) fixes 7, and (2497) sends 7 to 2. So in total 3 goes to 2. Thus far (19632...)

(376) fixes 2, (16) fixes 2, (64259) sends 2 to 5, and (2497) fixes 5. So in total 2 goes to 5. Thus far (196325...)

(376) fixes 5, (16) fixes 5, (64259) sends 5 to 9, and (2497) sends 9 to 7. So in total 5 goes to 7. Thus far (1963257)

The only number not permuted is 4, but we don't need to list it in this notation.

So yah, your answer looks good.