Permutationally invariant tomography has been tested experimentally for a four-qubit symmetric Dicke state, and also for a six-qubit symmetric Dicke in photons, and has been compared to full state tomography and compressed sensing. A simulation of permutationally invariant tomography shows that reconstruction of a positive semidefinite density matrix of 20 qubits from measured data is possible in a few minutes on a standard computer. The hybrid method combining permutationally invariant tomography and compressed sensing has also been tested.