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The reasons behind the gauge symmetry of the Standard Model,U(1)xSU(2)xSU(3), are still unsettled. One obvious feature is the lowdimensionality of all its subgroups. Under certain conditions, a negativeanswer to the question "why not larger groups like SU(15), or for that matter,SP(26) or E7?" is possible.  We have recently observed that fermions charged under large groups acquiremuch bigger dynamical masses, all things being equal at a high e.g. GUT scale,than ordinary quarks. Should such multicharged fermions exist, they are justtoo heavy to be observed today (and have either decayed early on if coupled tothe rest of the Standard Model, or become reliquial dark matter if uncoupled).Their mass scale is dictated by strong antiscreening of the running couplingfor those larger groups (with an appropriately small number of flavors)together with scaling properties of the Dyson-Schwinger equation for thefermion mass.  The generated fermion mass (assuming only few flavors, to avoid spoilingantiscreening) grows exponentially with the number of colors as M(N_c) ~exp(N_c) for scales much below the GUT scale. Large groups would be stronglycoupled already near the GUT scale and fermions charged thereunder havecorrespondingly large masses.

Spontaneous mass generation suggests small dimension of the SM group U(1)xSU(2)xSU(3)