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Our institution has a public IP address and now we want to create a private network using a switch which can connect to the Internet through this IP address. What class network should we use, and what would be the default gateway and DNS?

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    Network classes are dead, killed in 1993 by RFCs 1518 and 1519, which defined CIDR (Classless Inter-Domain Routing). Please let them rest in peace. Modern networking doesn't use network classes. – Ron Maupin May 17 '17 at 10:29
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You can use any of the addressing defined by RFC 1918:

10.0.0.0/8
172.16.0.0/12
192.168.0.0/16

for your private network. As far as the size of your private network, you can use whatever size fits your needs, and you can use multiple networks if necessary.

Using a switch will not work unless it is a layer-3 switch that can do NAT. Most switches cannot NAT, and you will probably need a router to NAT your private addressing to the public Internet.

You can use any usable host address in the network for your gateway address; it is simply the network address you assign to the gateway (the host, usually a router, that knows how to forward packets destined to a different network).

Your DNS is is handled by your DNS server. Some routers can have a built-in DNS server, you can have a stand-alone DNS server, or you can use one or more of the DNS servers available on the public Internet (recommendations are off-topic here).

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