I am supposed to label the vertices of a tetrahedron 1, 2, 3, and 4. I am supposed to explicitly describe the group of symmetries of the tetrahedron as permutations of these vertices. I know there are supposed to be 12 and I have written out all 12, see below. Then I am supposed to write the group table. I do this by composing the permutations, I think: a1.a1, a1.a2, a1.a3....a12.a11, a12.a12.

I don't know how I know that there are 12 permutations in the symmetric group. If 1, 2, 3, 4 are the possibilities how does one know that there are 12 elements in the symmetric group? In other words, if the 5 members in the set, how many elements are in the symmetric group? If there were 12? Also, I was asked to list the conjugacy classes. How do I find those? So far I have completed some of the table but I still have quite a ways to go.

Here are the 12 permutations I found:

a_{1}=(1)

a_{2}=(12)(34)

a_{3}=(13)(24)

a_{4}=(14)(23)

a_{5}=(123)

a_{6}=(243)

a_{7}=(142)

a_{8}=(134)

a_{9}=(132)

a_{10}=(143)

a_{11}=(243)

a_{12}=(124)