ℤ8 is generated by a single element, the square root of i, which I will denote by the letter a. Wherever a is sent, all other entries will be to the appropriate power of a. The eight elements will be called 1, a, i, ai, -1, -a, -i and -ai.
* 1 a i ai -1 -a -i -ai
1st | 1 1 1 1 1 1 1 1
2nd | 1 a i ai -1 -a -i -ai
3rd | 1 i -1 -i 1 i -1 -i
4th | 1 ai -i a -1 -ai i -a
5th | 1 -1 1 -1 1 -1 1 -1
6th | 1 -a i -ai -1 a -i ai
7th | 1 -i -1 i 1 -i -1 i
8th | 1 -ai -i -a -1 ai i a
Checking for orthogonality is a little tougher because you must turn one row into its complex conjugates before doing the dot product, but it does still work. This orthogonality makes group representation theory a useful part of physically describing an object whose group actions are represented by some set of matrices.
Tomorrow: The character tables for the quaterions and D4.
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