Linear representation theory of binary octahedral group

The binary octahedral group is a binary von Dyck group with parameters $$(4,3,2)$$, i.e., it has the presentation:

$$\langle a,b,c \mid a^4 = b^3 = c^2 = abc\rangle$$.

We denote the element $$a^4 = b^3 = c^2$$ as $$z$$. This element has order two.

This article discusses the linear representation theory of the binary octahedral group in characteristics other than 2 and 3.

Summary
Note that general linear group:GL(2,3) and the binary octahedral group are isoclinic groups of the same order. We know that isoclinic groups have same proportions of degrees of irreducible representations, therefore, in this case, the degrees of irreducible representations are the same for both groups. However, the character tables themselves are not identical. In fact, the fields generated by character values also differ from one another.

Degrees of irreducible representations
The degrees of irreducible representations can be computed using GAP's CharacterDegrees function:

gap> CharacterDegrees(SmallGroup(48,28)); [ [ 1, 2 ], [ 2, 3 ], [ 3, 2 ], [ 4, 1 ] ]

Character table
The character table can be computed using GAP's CharacterTable function:

gap> Irr(CharacterTable(SmallGroup(48,28))); [ Character( CharacterTable(  ),   [ 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable(  ),   [ 1, -1, 1, 1, 1, -1, 1, -1 ] ), Character( CharacterTable(  ),   [ 2, 0, -1, 2, 2, 0, -1, 0 ] ), Character( CharacterTable(  ),   [ 2, 0, -1, 0, -2, -E(8)+E(8)^3, 1, E(8)-E(8)^3 ] ), Character( CharacterTable(  ),   [ 2, 0, -1, 0, -2, E(8)-E(8)^3, 1, -E(8)+E(8)^3 ] ), Character( CharacterTable(  ),   [ 3, 1, 0, -1, 3, -1, 0, -1 ] ), Character( CharacterTable(  ),   [ 3, -1, 0, -1, 3, 1, 0, 1 ] ), Character( CharacterTable(  ),   [ 4, 0, 1, 0, -4, 0, -1, 0 ] ) ]

The character table can be displayed more elegantly using the Display function:

gap> Display(CharacterTable(SmallGroup(48,28))); CT1

2 4  2  1  3  4  3  1  3     3  1  .  1  .  1  .  1.

1a 4a 3a 4b 2a 8a 6a 8b 2P 1a 2a 3a 2a 1a 4b 3a 4b 3P 1a 4a 1a 4b 2a 8b 2a 8a 5P 1a 4a 3a 4b 2a 8b 6a 8a 7P 1a 4a 3a 4b 2a 8a 6a 8b

X.1    1  1  1  1  1  1  1  1 X.2    1 -1  1  1  1 -1  1 -1 X.3    2. -1 2  2  . -1  . X.4     2. -1 . -2  A  1 -A X.5    2. -1 . -2 -A  1  A X.6     3  1. -1 3 -1  . -1 X.7     3 -1. -1 3  1  .  1 X.8     4. 1 . -4  . -1.

A = -E(8)+E(8)^3 = -Sqrt(2) = -r2

Irreducible representations
The irreducible representations can be computed using GAP's IrreducibleRepresentations function:

gap> IrreducibleRepresentations(SmallGroup(48,28)); [ [ f1, f2, f3, f4, f5 ] -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], [ f1, f2, f3, f4, f5 ] -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], [ f1, f2, f3, f4, f5 ] -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, -1 ], [ 1, 0 ] ],     [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ]     ],  [ f1, f2, f3, f4, f5 ] -> [     [ [ -1/2*E(8)+E(8)^2+1/2*E(8)^3, -1+1/2*E(8)+1/2*E(8)^3 ], [ 1-1/2*E(8)-1/2*E(8)^3, 1/2*E(8)-E(8)^2-1/2*E(8)^3 ] ], [ [ -1-1/2*E(8)-1/2*E(8)^3, -1/2*E(8)-E(8)^2+1/2*E(8)^3 ], [ 1/2*E(8)-1/2*E(8)^3, 1/2*E(8)+1/2*E(8)^3 ] ], [ [ -1, -E(8)+E(8)^3 ], [ E(8)-E(8)^3, 1 ] ], [ [ E(8)+E(8)^3, E(4) ], [ -E(4), -E(8)-E(8)^3 ] ], [ [ -1, 0 ], [ 0, -1 ] ] ], [ f1, f2, f3, f4, f5 ] -> [     [ [ -1/2*E(8)+E(8)^2+1/2*E(8)^3, 1-1/2*E(8)-1/2*E(8)^3 ], [ -1+1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-E(8)^2-1/2*E(8)^3 ] ], [ [ 1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ], [ 1/2*E(8)+E(8)^2-1/2*E(8)^3, -1-1/2*E(8)-1/2*E(8)^3 ] ], [ [ 1, -E(8)+E(8)^3 ], [ E(8)-E(8)^3, -1 ] ], [ [ -E(8)-E(8)^3, E(4) ], [ -E(4), E(8)+E(8)^3 ] ], [ [ -1, 0 ], [ 0, -1 ] ] ], [ f1, f2, f3, f4, f5 ] -> [ [ [ 1, 1, 0 ], [ 0, -1, 0 ], [ 0, -1, 1 ] ], [ [ 1, 1, 0 ], [ 0, -1, 1 ], [ 0, -1, 0 ] ],     [ [ -1, 0, 0 ], [ 1, 0, 1 ], [ 1, 1, 0 ] ],      [ [ 0, -1, 1 ], [ -1, 0, -1 ], [ 0, 0, -1 ] ],      [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ],  [ f1, f2, f3, f4, f5 ] -> [ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 1, 1, -1 ] ], [ [ 0, -1, 1 ], [ -1, 0, 0 ], [ -1, -1, 0 ] ],     [ [ 0, 0, 1 ], [ 0, -1, 0 ], [ 1, 0, 0 ] ],      [ [ -1, 0, 0 ], [ 1, 1, -1 ], [ 0, 0, -1 ] ],      [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ],  [ f1, f2, f3, f4, f5 ] -> [ [ [ 0, 0, -E(3)^2, 1 ], [ 0, 0, 1, E(3)^2 ], [ -1, E(3), 0, 0 ], [ E(3), 1, 0, 0 ] ], [ [ E(3), 0, 0, 0 ], [ -E(3)^2, 1, 0, 0 ], [ 0, 0, -E(3), -1 ], [ 0, 0, E(3)^2, 0 ] ], [ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, E(3), -E(3)^2 ], [ 0, 0, -E(3)^2, -E(3) ] ], [ [ E(3), -E(3)^2, 0, 0 ], [ -E(3)^2, -E(3), 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ],     [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ]     ] ]