Public Groups are stored in the Group metadata type.
​
So in order to deploy a Public Group, you'd need to include Group metadata type in your metadata filter, which should look like like this:

You can also use Gearset's Default comparison metadata filter, which already includes the Group metadata type.

Below is an example of a Public Group found in the comparison results when Group metadata type is selected.

The user interface of the comparison results may differ slightly if you're using Compare 2.0 to retrieve a public group, in which case the interface would look as below:

For extra context, in this Salesforce org the group's location is in Users > Public Groups.

Some Gearset users have asked how to retrieve users assigned to a Public Group in their org.

For example, in this org an assigned use looks like this:

If this is the type of change you're looking to deploy between environments, we found out that Group metadata type won't retrieve users assigned to a specific Public Group.

It's because users are treated as data, not metadata. Therefore, metadata API won't support its retrieval.
​
So you'd need to consider adding it manually in your target org as a post-deployment step.