Is there a website that provides a free domain name? I wanted to make a domain for my group but really having a hard time cause of the payment

There are plenty of webhosting providers who include a free doamin name into their hosting packages www.siteground.com, www.**********.com, www.lunarpages.com etc.
I recommend you to read host's TOS before being tempted by any of this.

Chech web hosting section here. There were several providers posted by James.

Do you want to have it from the web hosting company or only a domain registrar?

Do you want to use it along with a package plan?

Domains cost the hosting company money. It is very unlikely that they will just give you a domain free of charge.

There are many hosting providers that will provide a free domain along with a hosting package.

In my opinion paid hosting is a lot more reliable too. Which will make your website even more successful.

Worth a shot...

*********** is one of the companies that gives a free domain name in the package which is registered in client's name.

No completely " free domain " available as domain need to pay cost itself. However many hosting provider provide free domain to you if you purchase hosting plan from them.

Registering a domain needs upfront cost to the provider (irrespective of what customer pays). So, unless the provider is running some kind of promotional offer, it would be almost impossible to get a free domain.