# Splitting fields of representations of generalized symmetric groups, 8

In this post, we give an example.

Let $G=C_3^8\, >\!\!\lhd \, S_8$ and let

$\pi = \theta_{\mu,\rho}=Ind_{G_\mu}^G(\mu\cdot \tilde{\rho}),$

where $\mu$ is a character of $C_3^8$ and $\rho$ is an irreducible representation of its stabilizer in $S_8$, $(S_8)_\mu$.

The real representations $\pi$ of $G$ are the ones for which

1. $\mu$ is represented by a character of the form

$(1,1,1,1,1,1,\omega,\omega^2) \ {\rm or}\ (1,1,...,1),$

and $\rho$ anything, or

2. $\mu$ is represented by a character of the form

$(1,1,1,1,\omega,\omega,\omega^2,\omega^2), \rho_1=(\pi_1,\pi_2,\pi_2)\in (S_4)^*\times (S_2)^*\times (S_2)^*,$

or

3. $\mu$ is represented by a character of the form

$(\omega,\omega,\omega,\omega,\omega^2,\omega^2,\omega^2,\omega^2), \rho_1=(\pi_2,\pi_2)\in (S_4)^*\times (S_4)^*,$

or

4. $\mu$ is represented by a character of the form

$(1,1,\omega,\omega,\omega,\omega^2,\omega^2,\omega^2), \rho_1=(\pi_1,\pi_2,\pi_2)\in (S_2)^*\times (S_3)^*\times (S_3)^*.$

The complex representations of $G$ are: the representations
whose characters have at least one complex value. Such representations $\pi = \theta_{\mu,\rho}$ are characterized by the fact that $(\mu,\rho)$ is inequivalent to $(\overline{\mu},\rho)$ under the obvious $S_8$-equivalence relation (which can be determined from the equivalence relation for representations in $G^*$).

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

There are no quaternionic representations of $G$.

The claims above follow from the fact that a representation
$\theta_{\rho,\mu}$ is complex if and only if $\mu$ is not self-dual.