  We give an explicit construction of a class of F-theory models with matter in
the three-index symmetric (4) representation of SU(2). This matter is realized
at codimension two loci in the F-theory base where the divisor carrying the
gauge group is singular; the associated Weierstrass model does not have the
form associated with a generic SU(2) Tate model. For 6D theories, the matter is
localized at a triple point singularity of arithmetic genus g=3 in the curve
supporting the SU(2) group. This is the first explicit realization of matter in
F-theory in a representation corresponding to a genus contribution greater than
one. The construction is realized by "unHiggsing" a model with a U(1) gauge
factor under which there is matter with charge q=3. The resulting SU(2) models
can be further unHiggsed to realize non-Abelian G_2xSU(2) models with more
conventional matter content or SU(2)^3 models with trifundamental matter. The
U(1) models used as the basis for this construction do not seem to have a
Weierstrass realization in the general form found by Morrison-Park, suggesting
that a generalization of that form may be needed to incorporate models with
arbitrary matter representations and gauge groups localized on singular
divisors.
