Your Facebook Page is incomparably more powerful than a group. The fact that FB allows Pages the same access to a user’s newsfeed as their friends is ENORMOUS. Every time you write a status update, post photos, or share a link…all your fans get it in their newsfeed. By becoming a fan, they have given your page permission to do this…and unless you over-expose them to your Page (twice a day updates?), they won’t hide you.
A group is relatively useless. All your activity on your group stays there, and your group members can’t know what’s happening unless they come to your group. And since they have so much incoming stuff in their newsfeed from Friends and Pages, they would rather Like/Comment that and not come to your group. 
