  In this paper, we construct an explicit minimal projective bimodule
resolution of a self-injective special biserial algebra $A_{T}$ ($T\geq0$)
whose Grothendieck group is of rank $4$. As a main result, we determine the
dimension of the Hochschild cohomology group ${\rm HH}^{i}(A_{T})$ of $A_{T}$
for $i\geq0$, completely. Moreover we give a presentation of the Hochschild
cohomology ring modulo nilpotence ${\rm HH}^{*}(A_{T})/\mathcal{N}_{A_{T}}$ of
$A_{T}$ by generators and relations in the case where $T=0$.
