• Each letter appears once and only once in each row and column.
  This means that we never have to go outside the eight starting letters to write down the inside letters of the table. This is called closure. The table for the reflections didn't have closure because, although we started with P,Q,R and S, the middle was made up of E,U,V and W.

• The transformation E leaves all the other transformations unchanged.
  E is called the identity transformation.

• Every transformation has a 'partner' or inverse so that together they give E.
  U has W and W has U and every other letter has itself.
  These other six transformations are called self-inverse.

  Little Note
  It's easier to describe the fourth property which this table has if I explain first that mathematicians have a shorthand way of writing combined transformations.
  They write U followed by P, for example, as PU.
  Notice the order here! This makes sense if you think of it as meaning PU(the square). The transformation of U is happening to the square first.

• Now, the fourth property of this table is that, if we do any three consecutive transformations, then we get the same result however we pair them off, provided we don't alter the order of the letters.
  For example, both U(VP) and (UV)P give the same result of R.

If a table has these properties then the elements which make it up form what mathematicians call a group.

Notice that the blue quarter tucked into the top right-hand corner of the table above is exactly the same as the first table which we made, which showed how the rotations combine together.
This table also fits the rules for a group and it is called a subgroup of the larger group.

..... and .....
Is it possible to make tables showing how their symmetries interact?