Thermal flow phenomena in a double lid–driven enclosure have many potential applications in numerous engineering domains. The present article theoretically investigates the heat transfer analysis of power-law fluid in a hexagonal cavity embedded with a square obstacle. The upper and lower lid walls of the cavity are considered heated along with the walls of the centered embedded square, while the rest of the cavity walls are thermally insulated. In addition, the horizontal lid walls of the cavity are uniformly moving in opposite horizontal directions. The mathematical modeling of the nonlinear fluid has been developed considering the continuity, momentum, and energy equations subjected to the appropriate boundary conditions. Due to the nonlinearity of the governing partial differential equations and of the viscosity models, we use the numerical scheme based on the finite element method The stable finite element pair (ℙ2/ℙ1) has been selected for the discretization purpose. The discretized nonlinear system is solved with the Newton method in conjunction with a direct linear solver in the inner iterations. The thermal flow features are exposed via streamlines and temperature contours for a set of governing parameters such as Reynolds number (Re), Prandtl number (Pr), and Grashof number (Gr) along with the power-law index (n). The values of local and average Nusselt numbers are calculated for the involved parameters. It is noted that the square obstacle has a strong impact for the formulation of streamlines and isotherms. Moreover, the power-law index has a strong impact on the values of the average Nusselt number and kinetic energy. The kinetic energy of the system increases with an increasing value of Gr and n, while Reynolds number has opposite effects on kinetic energy.