We develop a new four-node flat shell element which is accurate, efficient, and suitable to be used on general purpose. The new element has a hybrid Trefftz element with drilling degrees of freedom as a membrane part. We define the two independent displacement field: the internal displacement field that satisfies governing equations in the domain a priori and the boundary displacement field that is usually used as a conventional finite element method. The hybrid Trefftz variational formulation connects these two displacement fields on the boundary of the domain. To add drilling degrees of freedom, we introduce the Allman's quadratic displacement field to the boundary displacement field. As a result, our flat shell element has 6 degrees of freedom per a node. We also use the well-known DKMQ plate bending element for the plate part of the proposed element. The DKMQ element satisfies Mindlin-Reissner‘s plate theory along the edge of the element and gives proper behavior regardless of the thickness. A series of numerical experiments shows that the performance of the new element such as accuracy, rate of convergence, robustness to mesh quality, and so on.