Character Table for C22

I've been playing around with GAP, and have been studying the
character tables of Cyclic, Dihedral and Symmetric Groups. The
character tables are always square, where the number of conjugacy
classes always equals the number of irreducible expressions.

For cyclic groups, like C22, this is more obvious, because the
conjugacy classes are just the 22 transpositions of 22. I tried to
paste in the character table from GAP, but it looks terrible, so
here is just the second row, chi1, skipping the trivial character
chi0:

1 A B C D E -/E -/D -/C -/B -/A -1 -A -B -C -D -E /E /D /C /B /A

Here is the legend:

A = -E(11)^6
B = E(11)
C = -E(11)^7
D = E(11)^2
E = -E(11)^8

It looks like it is mapped on the Argand plane, so 11 steps
is -1, E(11) is a root, (2 steps), -E(11) would be 13 steps and the
others are based on

A) 13*6 mod 11=1
C) 13*7 mod 11=3
D) 2*2 mod 11=4
E) 13*8 mod 11=5

I am still learning all this, and trying to figure out why the mod is
11 instead of 22 etc. Also the smallest primitive root for 22 is 7,
not 13, but I may be confusing primitive roots with primitive roots
of unity here. I know, read the GAP manual. 7^phi(22) and 13^phi(22)
are 7^10 and 13^10 = 1 mod 22. 13^6=12 mod 22 and 1 mod 11. So 13
isn't even a 6th primitive root of unity! I know I must be missing
something in the cycle structure here, and need to brush up on
my complex analysis, at least. Any tips would be appreciated.

The character table for S22 is too big for GAP. But S12 is kind of
interesting, it is 77 X 77 with numerical values instead of the other
expressions. I know how to build S4's character table so I guess I
could learn this one. GAP also gives two tables before the main one
which are the centralizer orders based on the primes in the group,
and the next one has the order of the conjugacy classes, and then
a breakdown by primes (but only one prime, no powers, for each for
2,3,5,7,11). The order divides out unless the order of the conjugacy
class and the prime are relatively prime.

So are these tables of any value for studying musical systems?
And if so, would M24's character table also have some value? My little
find of the transpositional symmetry for D4, S3, C4XS3, D4XC3 and
D4XS3 is pretty modest, but I hope it will lead to other things.

Sorry to ramble, I will work to tighten up my ideas, but I know
you're all fast readers.

Paul Hj