In this post, we give an example.

Let and let

where is a character of and is an irreducible representation of its stabilizer in , .

The real representations of are the ones for which

- is represented by a character of the form
and anything, or

- is represented by a character of the form
or

- is represented by a character of the form
or

- is represented by a character of the form

The complex representations of are: the representations

whose characters have at least one complex value. Such representations are characterized by the fact that is inequivalent to under the obvious $S_8$-equivalence relation (which can be determined from the equivalence relation for representations in ).

The complex representations of are the remaining representations not included in the above list.

There are no quaternionic representations of .

The claims above follow from the fact that a representation

is complex if and only if is not self-dual.