Is it reasonable for a professor to ask students to stay 30-45 minutes past scheduled class time every week?

I guess I'm going to be the contrary opinion here. I don't think your professor's request is reasonable at all. Class time is scheduled the way it is so that people can plan the rest of their lives around the schedule. In this case, nobody but you seems to have anything from 12:50 on, but that shouldn't encourage the prof to monopolize that time.

Some may think your extracurricular activity is trivial or that you should skip it, but that math club could easily have been another class or your job. The professor is in a position of power, and he should respect his position by not trying to compel you to work outside the system because it's convenient to him. Several of your peers may actually appreciate that you're the lone hold out even if they won't ever admit it.

I would have no problem with the prof offering extra sessions where no required material was covered. This could be extra practice working example problems or a Q&A session to clear up anything confusing from class. That being said, I think professors should be held to teaching the required stuff in the allotted time.

This is a strange situation: three out of the four answers so far express what is in my experience a very nonstandard view. Admittedly Bill Barth's answer is by far the most highly upvoted. But I think it is worth recording another opinion.

First, as for so many questions on this site, context matters. It sounds to me like the OP is an undergraduate at a North American college or university. My answer will apply to that.

At a university (and at all but the smallest colleges), scheduling classes is a major production. It is done at least a semester and often a year (or more) in advance, and a lot of time and effort go into crafting a schedule which makes the best possible use of limited resources: classrooms, peak hours, people's time, and so forth. Moreover the total amount of class time is also fixed in advance and has an official status as the number of credit hours for the course. The number of credit hours is meant to be correlated to the amount of material covered. In practice this means little in an absolute sense and one course can certainly be much more or less ambitious than another, but carries a meaning across different sections of the same course and within the course offerings of a given department.

Am I right in being outraged?

Outraged is a strong word, but the OP is right to disapprove of the practice and perfectly within his rights to object to it. I do not find the professor's request reasonable. In fact I find it very surprising that in an entire class of students, only one other student has an extra 30-45 minutes in their schedule on Friday afternoons at 1 pm: this is typically "peak class time". (I wonder if there is some missing context here: e.g. maybe the other students form a cohort who are all taking the same courses scheduled at the same time. That makes some difference, but not enough to change my answer.)

The OP has made his schedule for the entire semester in advance and has signed up for the class assuming that it will meet at a certain time, which was fixed by the registrar in advance. He has a legitimate conflict with staying an extra 30-45 minutes. This conflict is academic in nature and it involves students other than those taking the professor's course. (In my experience with academic culture, it is obnoxious to try to get someone else's academic event rescheduled because you want to change the schedule of your own academic event. It is in the nature of a university that all of its participants are engaged in multiple events. We schedule things well in advance precisely to avoid conflicts like this.) However, even without a specific academic conflict the OP should feel free to veto the request: if e.g. he has made a schedule where he leaves campus at 1 pm, that would be reason enough to object.

Is it not the professor's job to teach the necessary course material in the time allotted, as opposed to having students stay late for no credit?

Yes, it is! I agree precisely. Something is not adding up here: the course is a standard, required one for the major, so it has presumably been given many times before by other instructors and there is a de facto standard expectation for its content, as detailed in e.g. an official course description. Of course the instructor of a course should have some leeway in the choice of topics and even the intensity of the course....but not complete leeway. This instructor is proposing to increase the class time by about 25%: why is this necessary? If he is covering 25% more material than is standard, that's potentially problematic. If he takes 25% more time to cover the material than is standard, that is also problematic. Again some context is lacking, but as described the instructor seems to be having some serious problems "coloring between the lines". The OP should not have to adjust his schedule on account of this.

I honestly think that almost any higher up university official -- the department head, the registrar, a dean -- would be chagrined to hear about what the instructor is proposing. At least that's the case in the university culture I'm familiar with. If the OP has a faculty contact (in any department) he is comfortable talking informally with, I would recommend broaching the issue and seeing what the faculty member has to say.

Courses and their timings will usually be organised so that students having to attend to multiple related courses can attend to all of them. Of course only the official time is organised that way, unofficial time can't be organised.

So what if some maths professor decides that his course on Friday from 11am to 11:50am should be an hour longer? Now the professor running the circuits class would be outraged if everyone comes late to his course.