It all depends on how your router is configured to assign IP address. With DCHP enabled, your router will have a default gateway of what ever you set it at, or whatever the default is (usually the 192.168.1.1 you're referring to) and will assign subnets within that parameter. What ever your public IP will be is determined by your ISP which is hardly ever static (unless you ask for and pay more for a static IP), they issue IP's with leases that expire and will change time to time. If you ever want to change your IP, power cycle your modem. Sometimes it will issue the same one you had before, other times it will give you a new one.
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