1. Calculate $6^5$. This means multiplying 6 by itself 5 times: $$6^5 = 6 \times 6 \times 6 \times 6 \times 6.$$\n2. Compute step-by-step: $$6 \times 6 = 36,$$ $$36 \times 6 = 216,$$ $$216 \times 6 = 1296,$$ $$1296 \times 6 = 7776.$$ So, $$6^5 = 7776.$$\n\n3. Now calculate $3.4 \times 10^2 \times 4 \times 10^4$. Group the numbers and powers of ten:\n$$ (3.4 \times 4) \times (10^2 \times 10^4) = 13.6 \times 10^{2+4} = 13.6 \times 10^6.$$\n\n4. Review the answer choices for the expression $3.4 \times 10^2 \times 4 \times 10^4$:\n- A. $3.8 \times 10^8$\n- B. $13.6 \times 10^8$\n- C. $13.6 \times 10^6$\n- D. $3.8 \times 10^6$\n\n5. The correct choice corresponds to the calculated result, which is choice C: $$13.6 \times 10^6.$$