The planar algebra of group-type subfactors

If G is a countable, discrete group generated by two finite subgroups H and K and P is a II1 factor with an outer G-action, one can construct the group-type subfactor PH⊂P⋊K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones' planar operad is determined explicitly.