If you are using winbind to authenticate with your AD domain, then you can use the wbinfo command to get this information:
To list all domain groups:
$ wbinfo --domain-groups unix_group1 unix_group2 DOM2+windows_group1 DOM2+windows_group2 DOM3+windows_group3
This option will list all groups available in the Windows domain for which your host is operating in. Some groups are listed with the preceeding domain name (DOM2+). The ones without a domain name are from the local domain.
To establish the hosts own domain to which it belongs:
$ wbinfo --own-domain DOM1
To list all domains that the hosts samba daemon is operating in:
$ wbinfo --domain-groups DOM1 DOM2 DOM3
To confirm the group membership, you can set the domain to search in then check for domain groups:
$ wbinfo --domain DOM1 --domain-groups | grep unix unix_group1 unix_group2