How to automatically turn something off at 11pm and on at 6am?
It is likely that your router can be plugged into an ordinary power strip. There are commonly available power strips with seven day digital timers built-in. The SE is not a product recommendation forum, which means it's not proper for me to provide specific product identifiers, but you can use your preferred search engine using the terms "digital timer power strip" and select the item which best meets your needs.

I have a pair of these devices in use, as the day of timed operation varies from week to week in my application. For your daily use, you may be able to locate a 24 hour version rather than a full-week version, making the programming slightly easier.