The quaternions are a number system in the form + + + . The quaternions ±1, ± , ± , ± form a non-abelian group of order eight called quaternion group. Quaternion group can be represented as a subgroup of the general linear group 2 (C). In this paper, we discuss some group properties of representation of quaternion group related to Hamiltonian group, solvable group, nilpotent group, and metacyclic group.