Let $G$ be a finite group and $\phi\colon G\to \mathrm{GL}_d(\mathbb C)$ be an irreducible representation, with character $\chi$. Recall that

- $\phi$ is
*complex type*if $\chi$ is not real-valued, - $\phi$ is
*real type*if $\phi$ is the complexification of a representation $G\to\mathrm{GL}_d(\mathbb R)$, - $\phi$ is
*quaternionic type*otherwise, i.e. $\chi$ is real-valued but $\phi$ is not real.

(Equivalently, $\phi$ is quaternionic if $\mathbb{C}^d$ has a $G$-invariant symplectic form, or if there is a $G$-equivariant, conjugate-linear operator on $\mathbb{C}^d$ whose square is $-\mathrm{id}$.)

**Question**: Is there an example of a finite group $G$ of order $4k+2$ for some $k$, such that $G$ has a quaternionic irreducible representation?

Some thoughts that may be relevant:

- The Frobenius--Schur indicator $\frac{1}{|G|}\sum_{g\in G}\chi(g^2)$ is $-1$ iff $\phi$ is quaternionic, $+1$ iff $\phi$ is real, and $0$ iff $\phi$ is of complex type.
- A group of order $4k+2$ must decompose as the semidirect product of a cyclic group of order two acting on a group of order $2k+1$.
- Every nontrivial irreducible representation of a group of odd order is automatically of complex type (maybe this is a theorem of Burnside).
- The degree $d$ has to be even, since $\mathbb{C}^d$ must have a symplectic form.
- A counting argument using the Frobenius--Schur indicator, and a count of the the number of elements of order 2, shows that the degree $d$ must be strictly greater than $2$. So we need an irreducible representation of degree at least $6$.
- Searching on GAP didn't give any examples for $|G|\leq 150$.

This isn't really a research question, it's just something curious. I'm asking for a friend.