  We show that higher order inter-group covariances involving even number of
qubits are necessarily positive semidefinite for N qubit separable states,
which are completely symmetric under permutations of the qubits. This
identification leads to a family of sufficient conditions of inseparability
based on the negativity of 2kth order inter-group covariance matrices of
symmetric N-qubit systems. These conditions have a simple structure and detect
entanglement in all even partitions of the symmetric multiqubit system. The
observables involved are feasible experimental quantities and do not demand
full state determination through quantum state tomography.
