Is it possible to manage a free public google group with terraform?

Question

I manage several email groups for an open-source project using the free version of Google Groups. The groups were created using a personal gmail.com account. I'd like to manage these groups with terraform.

https://support.google.com/groups/answer/46601?hl=en:

There are two versions of Groups—the free version for everyone, and a version for organizations (work, school, and so on). The tasks you can do and features you can access depend on which version you have. If you're signed in with a work or school account, the tasks and features available to you are customized by your administrator.

I know how to manage organization (Google Workspace) groups with terraform using the hashicorp/google-beta provider. However, I have not figured out how to do so with free groups, or even whether it's possible.

For an organization (Google Workspace), a group may be created like this:

resource "google_cloud_identity_group" "my_group" {
  provider     = google-beta
  display_name = "My Example Group"
  parent       = "customers/C0..."

parent is the id of the parent organization. I have no idea what would go there for a public group created with a gmail.com account. (The value must be set.)

Again, this question is specifically whether it's possible (and how) to use terraform to manage a free (non-workspace) google group that was created with a gmail.com account.