Ok so i get how something like a rotation group or some subgroup of it or even a general linear group of transformations represented as a 3x3 matrix can have an action on R3 vectors. Im also aware than the abstract rotation group can have a representation as a matrix in any size. But how does the opposite work? How would some big abstract group like "E8" or " the monster" or SU(9)xSpin(17) have an action on some euclidean R3 vector?
Like would that look like a set of 3x3 matrixes that have a concrete product with a vector?