## The Character Table of S_3xZ_3

**Point of post: **In this post we use our recently developed theory on the irreducible representations (characters) of the products of groups to find the character table of a more formidable group, namely .

**Motivation**

Let’s put our newly developed theory to work.

*Character Table of *

We begin by constructing the character tables for and . The first of these we have already done and is given by

and the second one is fairly easy. Namely, we know that the number of non-equivalent irreducible characters of is . And it’s easily verifiable that the three characters and where and . Thus, the character table is given by

From our previous theorem we may conclude that the representatives from the equivalency classes of are

and the irreducible characters are

and thus the character table for , using our characterization of character tables of product groups

We therefore ascertain using our results about character tables that . Moreover, the normal subgroups of are , and .

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