Several very useful formulas, which give the cycle indices of the binary operation of the sum, product, composition and power group of M and H in terms of cycle indices of M and H (Harrary 1967) have been discussed. One very useful binary operation on groups, which has not been exploited, is the semidirect product. The question is: How can we express the cycle index of G in terms of the cycle indices of M and H, G a semidirect product of M and H? This book answers this question by considering the cycle indices of some particularly important semidirect product groups; namely, the dihedral groups, the Frobenious groups and a special case of semidirect products-the internal direct products.