Jim Caprioli at The Finland Station

There is a unique nonabelian group of order 21, which can be realized as the group of affine transformations x |--> ax + b of the line over the field with seven elements, with a a cube root of unity in that field. Find the irreducible representations and character table for this group.

---

I thought the only groups of order 21 were C_21 and C_3 X C_7 which are abelian. Since there must be at least one non-abelian group of order 21 I presume it is some semi-direct product. My recipe for this exercise is as follows:
- construct group;
- determine # conjugacy classes;
- calculate # linear characters = | Group : DerivedSubgroup |;
- calculate the dimensions of the required irreducible representations;
Do Until all reps found
- find representation;
- calculate character;
Od;
- find 21 matrices representing the group (not required for 3.26*)

Status>
- Group constructed -and- analyzed;
- Working on finding the irr reps.