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This book covers the construction of irreducible characters of the 2 by 2 and 3 by 3 special linear group over the integers modulo a power of an odd prime. In the 2 by 2 case a construction of every irreducible character of this classical group is given, without calculating the character values. The first chapter starts off with a brief introduction into this group and introduces all the necessary character theory. To produce each irreducible character a normal subgroup of our group is found and cleverly mapped into the complex numbers. Clifford Theory is then used to find the irreducible character of interest. After counting all the characters of each degree and comparing them to the number of conjugacy classes it is shown that all the irreducible characters have been found. The last chapter shows that this method can extend to the 3 by 3 case. Throughout the book many details are given to provide clear and concise reasoning.