God has placed in a body, a community, with all kinds of people. Some we have affinity with, some we don't. Yet we are called to love all of them. In Community Groups we gather together with everyone in our church from a certain geographic area, regardless of stage of life, age or family size, etc. Here we learn to love those who are different from us, in His name. As we lay down our lives and love others, they are blessed, and we are too. More than this, Community Groups then become little outposts of heaven, displaying the love of God to our neighborhoods.
While Community Groups gather strictly according to geography and are therefore larger, Discipleship Groups are smaller and gather according to affinity. Some study the Bible or read a book; others gather to ride motorcycles or bicycles; others discuss the prior week's sermon. In all of them, we seek to, as the Puritans put it, "attend to one another's souls" with the gospel. But the purpose is not just for us - it's for our neighbors too. In DG's we seek to "get on mission" together. Here we learn to love God more and love our neighbors, by sharing the gospel with them. In DG's we help each other obey both the Great Commandment and the Great Commission - and have some fun doing it!
These groups are not "core" groups, but are useful for connecting to others.