# Degrees of irreducible representations of direct product are pairwise products of degrees of irreducible representations of direct factors

From Groupprops

## Contents

## Statement for two groups

### Over any field

Suppose and are finite groups. Suppose is a field. Suppose are the degrees of irreducible representations (possibly with repetitions) of over and are the degrees of irreducible representations of .

Then, the degrees of irreducible representations of over are:

### Over a splitting field

If we take to be a splitting field for *both* and , then the above relates the degrees of irreducible representations for over a splitting field. For instance, we could take to be or . Note that degrees of irreducible representations are the same for all splitting fields (as long as the characteristic does not divide the group order), so this relates universal lists of numbers for .

## Statement for multiple groups

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## Related facts

- Conjugacy class sizes of direct product are pairwise products of conjugacy class sizes of direct factors
- Number of conjugacy classes in a direct product is the product of the number of conjugacy classes in each factor