Jim Caprioli at The Finland Station

What have we got sofar? Four characters X1, X2, X3, X4 ( see below ).
What are they? The first row represents an element in the original group, as constructed in Mathematica. The second row is an element of the free group which is isomorphic to the previous one. The rows beginning with X are the characters.
How are they calculated? The numbers on each row are values of morphisms from Z4x|Z4 -> { (-1,1), * } There are four morphisms as such which preserve the structure of Z4x|Z4 and are thus homomorphisms. The homomorphisms are well tested in Mathematica.
- X1 = FormMorphoid[Mod[#[[2]],1]+1&, G, FormGroupoid[{1},Times]]
- X2 = FormMorphoid[(-1)^#[[1]]&, G, FormGroupoid[{1,-1},Times]]
- X3 = FormMorphoid[(-1)^#[[2]]&, G, FormGroupoid[{1,-1},Times]]
- X4 = FormMorphoid[(-1)^#[[1]]*(-1)^#[[2]]&, G, FormGroupoid[{1,-1},Times]]
Doing this exercise helps building a 'have been there, done that' attitude which is just as important in math than anywhere else, I suppose...
Notice the symmetry in the table below (2nd part, cols5-10). Enjoy! Working on this stuff is close to euphoria.
(To be continued...)

cols 1-4
	1	2	3	4
	0 0	0 2	2 0	2 2
	e	a2	b2	a2b2
X.1	1	1	1	1
X.2	1	1	1	1
X.3	1	1	1	1
X.4	1	1	1	1

cols 5-10 
	0 1 	1 0 	1 1 	2 1 	3 0 	3 1
	a	b	ab	ab2	b3	b3a
X.1	1	1	1	1	1	1
X.2	1	-1	-1	1	-1	-1
X.3	-1	1	-1	-1	1	-1
X.4	-1	-1	1	-1	-1	1