Symmetric extensions of dihedral quandles and triple points of non-orientable surfaces

Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. e smallest example R ˜ 3 of order 6 that is an extension of the three-element dihedral quandle R 3 , various symmetric quandle homology groups are computed, and applications to the minimal triple point number of surface-knots are given.