So there is a big group based on the fandom of my little pony
My Little Pony is based on the toy franchise based on dolls and play sets. It aired on Discovery Family. What is the name of this fan group?
Friendship is Magic.  It is actually mostly for adults and cosplayers
It was a tv series based on My Little Pony and it also aired on Discovery Family. Do they have events and get togethers?
They have conventions.  One of the major ones is called BronyCon.  
So BronyCon meets annually in the United States for fans of the show.