. . Assume all of the following are true: G is a group of order 24. H is a subgroup of G isomorphic to D3, with generators r and f. (This is the group of symmetries of the equilateral triangle.) G is generated by elements r and f along with a third element c, which has order 4. You should be able to identify three distinct ways this can happen, depending upon the nature of H's containment within G. Demonstrate an example of each, via a partial Cayley diagram of G emphasizing H. Each of your three diagrams should show enough to convey that the the bulleted items are true, along with your claim about how H is contained within G. Your diagram does not need to be a FULL Cayley diagram of G, because filling out all of the details and knowing that what you are left with is in fact an actual group, is very very tricky. If you have followed along with our visualization techniques so far, you should know what details are important and need to be shown. Remember that in a Cayley diagram (even a partial one), arrows for different generators need to be distinguishable, and should show a direction (unless they are their own inverse). . . Assume all of the following are true: G is a group of order 24. H is a subgroup of G isomorphic to D3, with generators r and f. (This is the group of symmetries of the equilateral triangle.) G is generated by elements r and f along with a third element c, which has order 4. You should be able to identify three distinct ways this can happen, depending upon the nature of H's containment within G. Demonstrate an example of each, via a partial Cayley diagram of G emphasizing H. Each of your three diagrams should show enough to convey that the the bulleted items are true, along with your claim about how H is contained within G. Your diagram does not need to be a FULL Cayley diagram of G, because filling out all of the details and knowing that what you are left with is in fact an actual group, is very very tricky. If you have followed along with our visualization techniques so far, you should know what details are important and need to be shown. Remember that in a Cayley diagram (even a partial one), arrows for different generators need to be distinguishable, and should show a direction (unless they are their own inverse)