I assume that you mean suspended without pay.
First, look at the company's disciplinary and attendance policies. Regardless of whether you are salaried or hourly-paid, unionized or not, I would be surprised if six-week suspensions are an option at all. My opinion is that you're in an unemployment no man's land. Although technically you can return to work after six weeks, your employer is betting you will just find another job and not file for unemployment. Most people who work need the money and benefits that come with the job, and that is a long time to be out of work.
The longest disciplinary suspension I've known is one month, else the employee was discharged. Taking such a drastic disciplinary measure of six weeks makes me think it might not have been the first incident for you. (The companies I've dealt with discharge employees for no call/no shows rather than suspend them.) There's a point at which lengthy disciplinary suspension is really unemployment. Therefore, you could try filing for state unemployment ASAP and/or make a complaint to your state's wage and hour division. An attorney is not required to file such a complaint, but an attorney can let you know the merits of your case and your options and thus boost your chance of success.
Second, only you truly know for sure whether you failed to call/failed to show. Companies typically have very specific protocols for calling off work, so it would be very useful to you if you could simply demonstrate compliance with those rules. Your phone records might help as well as who you talked to and any witnesses. If you're in the wrong, a more compelling argument will mix regret with a complaint that the punishment is overly harsh.
Third, if your company has any policy, they need to follow it. It's generally better to have no policy than to have one and not follow it consistently. Therefore, do you know anyone who had a no call/no show with the same circumstances and was treated differently? If so, list these out. Has anyone else at your workplace been suspended this long or are you the only one? Consider whether there might be an underlying motivation of some kind for being so harsh with you.
Fourth, have you already tried to appeal the suspension within the company?
I hope I've given you some ideas. Six weeks seems like pounding a gnat with a hammer, although you certainly need to be more dependable.