Abstract: A group is D-quasirandom if all its non-trivial unitary representations have dimensions more than D. This property is obviously not definable in first order logic, and in particular, an ultraproduct of quasirandom groups will in general fail to be quasirandom. In this talk, I shall present the covering properties, which is definable in first order logic, and shall characterize the quasirandomness to a certain degree. A group is said to have a good covering properties iff it has an element g, and the conjugacy classes of all small powers of g are fast expanding. These properties will be almost equivalent to quasirandomness if we ignore the cosocle of a group (the intersection of all maximal normal subgroups). Furthermore, it is preserved under arbitrary products and quotients. We shall also discuss its connections to ultraproduct of quasirandom groups, Bohr compactifications and ergodic theory results.