Jim Caprioli at The Finland Station

I continued to work on "Z4:Z4", one of the "unnamed groups of order 16" (is that a fact?). Under the tab "notes" in the program Finite Group Behavior (see previous note) the author explains the group as follows:
"...
This is a semidirect product of Z4 by Z4.
First we define a homomorphism P: Z4 -> Aut(Z4) by the rule that
P(1)=P(3) are the functions (0->0,1->3,2->2,3->1) and P(0)=P(2) are the identity.
The group operation is thus ab*xy=[a+P(b)(x)](b+y) where + denotes addition in Z4.
..."

The construction of the group in Mathematica resulted in elements of pairs of numbers. (0,0), (0,1), etc. I wanted however, to create a multiplicative group with single elements, like, a, b. So I had to find some morphism from the group I already created to the desired group. GAP is better suited to create groups from elements and relations, a so called free group. The library of FGB contains all groups upto order 40. These groups can be completely analyzed, although the program doesn't do Character Table stuff like GAP. I analyzed the Cayley Table of Z4:Z4, I concluded that Z4:Z4 has two generators and three relations.
Generators: a and b
Relations a^4=1, b^4=1 and b*a=a^3*b.
The morphism is then as follows.
(0,0) -> 1
(0,1) -> a
(0,2) -> a^2
(0,3) -> a^3
(1,0) -> b
(1,1) -> ab
(1,2) -> a^2b
(1,3) -> a^3b
(2,0) -> b2
(2,1) -> ab^2
(2,2) -> a^2b^2
(2,3) -> a^3b^2
(3,0) -> b^3
(3,1) -> ab^3
(3,2) -> a^2b^3
(3,3) -> a^3b^3
Let's verify this in GAP

gap> f:=FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> a:=f.1;
f1
gap> b:=f.2;
f2
gap> G:=f/[a^4,b^4,b^-1*a^3*b*a^-1];
<fp group on the generators [ f1, f2 ]>
gap> Size(G);
16
gap> Display(G);
generators = [ f1, f2 ]
relators = [
f1^4,
f2^4,
f2^-1*f1^3*f2*f1^-1 ]
Looks OK.

But characters?!?! Nope. No, Nada yet. I hope to produce some results by tomorrow. If I am not able to come up with the characters on this one then I should take another group and do it all again.