HI Nayla McCarty -

You can manually assign a ticket to a Group in the Assignee field -- just select the Group without also selecting an assignee: Manually assigning a ticket

The Group field has a limitation that it can't be set directly by end-users, although you can allow them to see a ticket's group assignment when they're looking at the ticket in the My Activities section of your Help Center (if you have that enabled). To do that, you'd edit the Group ticket field and select "Customers can view" under Permissions. See Editing ticket fields

That said, you can create a custom drop-down field with values that correspond to your various groups (be sure to set the field's permissions to "Customer can edit"), add that to your ticket form, and then create a Trigger for each field value, so that when the ticket is created, if the field value matches a specific group, the ticket is assigned to that group. The conditions and actions for such a trigger might look like this (and again, you'd need a separate trigger for each value of the field):

For more information on creating custom fields, adding fields to ticket forms, and on creating triggers, see Adding custom fields to your tickets and support request form and Triggers resources

Hope that helps!