Can you start with the complex character table. The power maps will let you see 
which conjugacy classes have to be combined and then once you’ve done that, 
characters which have become identical also need to be merged,



> On 29 Jun 2019, at 02:11, Vince Giambalvo <vinc...@hawaii.edu> wrote:
> 
> Dear Gap Forum,
> Does anyone have thoughts on how to construct real character tables for small 
> finite groups. It seems as if this is
> something that someone has already done a long time ago. I suppose it is 
> possible to construct the “real” conjugacy 
> classes (those containing both an element and its inverse), but if someone 
> has an easier suggestion or has already done it, 
> I would appreciate the help.
> 
> Thanks in advance.
>     Vince
> 
> 
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