This exercise is about putting Cayley's Theorem into practice. First, read and study the theorem. Realize that this result by itself is primarily of theoretical interest, but with some more theory we could get into some subtler aspects of this (a subject known as ``representation theory'').
You should create these representations mostly with pencil-and-paper work, using Sage as a fancy calculator and assistant. You do not need to include all these computations in your worksheet. Build the requested group representations and then include enough verifications in Sage to prove that your representation correctly represents the group.
Begin by building a permutation representation of the quaternions, Q. There are eight elements in Q (±1,±I,±J,±K), so you will be constructing permutations of a set of 8 elements.