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, I found that the McLaughlin 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 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)?