is there bus service from Chennai to Chidambaram or Kumbakonam? What is the travel time?

Yeah! From Chennai, there are so many buses to Chidambaram & Kumbakonam. Govenment buses like State Express Transport Corporation (SETC), Tamil Nadu State Transport Corporation (TNSTC) & Omni buses like Rathimeena, Universal, KPN are there to both chidambaram & kumbakonam. The travelling time for chidambaram via GST is 5 hours & by ECR is 4.5 hours. Then the travelling time for kumbakonam is 7 hours.
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