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I would be very surprised if the mathematicians have not already investigated and named it, so I am looking for two things :

a) A resource/contact by which I can find the name of the group, given its algebraic root vectors, and from that of course anything that is already known about it by other researchers.

b) Help from a physicist who knows how to make a formal comparison between the numbers in the standard model and the properties of a proposed group for a GUT.

I have the equation that generates the set of root vectors for the group, and of course the generated set of root vectors themselves, but that is all the knowledge I have about the group. It seems to have some similarities to the SO and Spin groups. Curiously the equation seems to be a rather oddball way of generalising the Lorentz transform of special relativity.

There are some pretty neat symmetries in this group, and it has already led a few years back to a proposed new dimensional grouping of physical constants from which a more accurate value of G (Newtons's constant) can be calculated (G = 6.673870 92(30) x10-11 m3kg-1s-2 using CODATA 2010 values). The value is holding up very well to new more accurate experiments of G published last year.

There are also internal symmetries that should relate the standard model to dark matter, indicating an exact 1/6 visible 5/6 dark ratio that remains in the empirical bounds of the latest CMB results. This six fold symmetry should I suspect show up as a resonance at exactly 6 times the observed Higgs energy at CERN. Finally this group would indicate dark matter should interact not only gravitationally, but with some residual weak force effects with visible matter, but no electromagnetic or strong force interactions. But then my understanding of this group remains far from complete, and all this could be a crock of the proverbial.