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.



Can groups of twice-odd order have quaternionic representations?
I'm not sure how acceptable it is to give answers referring to theorems in textbooks. If you'd like me to reproduce the relevant argument, then please let me know.


Chapter III.11 of Simon, *Representations of finite and compact groups*, shows the following (in the notation there): if $H \le G$ with $H$ finite and $[G : H] = 2$, then (Theorem 1) irreducible representations of $G$ are in bijection with orbits of irreducible representations of $H$ under the action of a conjugator $y \in G \setminus H$; and (Theorems 2 and 3), if (either) representation $V$ of $H$ in an orbit is complex, then the corresponding representation of $G$ is quaternionic only if $V(y^2) = -1$. However, this is impossible if $H$ has odd order.


