The new quilt's math

This is what the new quilt will look like when it is done!  Is the multiplication table for the group of two by two matrices with entries in the field with three elements and determinant 1.  We really like this as a companion to the symmetric group quilt since both groups have twenty four elements, but they have very different subgroups!  The symmetric group has a subgroup of order 12 (the alternating) group and if you look at the picture above you can see that the yellow and orange squares in the top left corner are a subgroup of order eight.  We also have a subgroup of order 4 in this quilt (the yellow squares in the top left corner.)

You can also see that the eight by eight squares of yellow and orange, red and purple, and blue and green are always together.  This means that the yellow/orange subgroup is normal.  The four by four squares of different shades of the same color don't have this nice property and it means that it is not a normal subgroup.  If you look really carefully you can see that the two by two subgroup in the very top left is normal!