I'm writing a software package to decompose group representations, and am struggling to find good examples of quaternionic-type representations of dimension > 4.


Reading [MathOverflow](https://mathoverflow.net/questions/54800/why-are-there-so-few-quaternionic-representations-of-simple-groups), I found that the [McLaughlin group](https://en.wikipedia.org/wiki/McLaughlin_sporadic_group) has quaternionic representations of dimension 3520 and 4752. This is too big for my numerical experiments. Note that the aforementioned MathOverflow questions has links to two papers that I can't easily grasp. 


The [other answer I found](https://mathoverflow.net/questions/46900/are-there-real-vs-quaternionic-conjugacy-classes-in-finite-groups) describes a group where quaternionic/symplectic representations have dimension 4.


Are there finite groups with quaternionic type irreducible real representations of dimension > 4 and < 1000 (say)?



Which dimensions exist for irreducible quaternionic-type real representations of finite groups?
Let $\mathbf{H}$ be the skew field of real quaternions. Let $Q\subset \mathbf{H}^\*$ be the quaternion subgroup of order 8, namely $Q=\{\pm 1,\pm i,\pm j,\pm k\}$.


Let the wreath product $G\_n=\mathfrak{S}\_n\ltimes Q^n$ (of order $n!8^n$) act on $\mathbf{H}^n$ (viewed as right $\mathbf{H}$-module) as monomial matrices (with nonzero entries in $Q$), namely $Q^n$ acts diagonally: $(q\_1,\dots,q\_n)\cdot (v\_1,\dots v\_n)=(q\_1v\_1,\dots,q\_nv\_n)$, while $\mathfrak{S}\_n$ permutes the coordinates.


Claim: this representation on the real $4n$-dimensional vector space $\mathbf{H}^n$ is irreducible (for $n\ge 1$). Indeed, starting from any nonzero vector $v$, perform a permutation to ensure $v\_1=1$, then act by $(-1,0,\dots,0)$ and subtract, to get $v\_1\neq 0$ and $v\_i=0$ for $i\ge 2$, and finally multiply by elements of $Q$ and use permutations to generate all other vectors.


Then, this can be viewed as an irreducible complex $2n$-dimensional representation, of quaternionic type (and conversely any irreducible complex representation of quaternionic type has positive even dimension).


Actually, in the above argument, $\mathfrak{S}\_n$ can be replaced with any transitive subgroup (for the irreducibility argument to hold), and hence we can replace $G\_n$ with a subgroup of order $n8^n$.


