Determining eigenvalues, determinants, and inverse for a general matrix is computationally hard work, especially when the size of the matrix is large enough. But, if the matrix has a special type of entry, then there is an opportunity to make it much easier by giving its explicit formulation. In this article, we derive explicit formulas for determining eigenvalues, determinants, and inverses of circulant matrices with entries in the first row of those matrices in any formation of a sequence of numbers. The main method of our study is exploiting the circulant property of the matrix and associating it with cyclic group theory to get the results of the formulation. In every discussion of those concepts, we also present some computation remarks.