If I can help, then it isn't a waste of my time man I'm glad to help.
I understand exactly where you're coming from. I struggled with approaching for the longest time, especially in the daytime, and in a classroom setting everything you do tends to be under a sort of magnifying glass. However, reading over your posts again, I would also like to know how you came to like this girl if you have never spoken to her. Purely physical attraction? Careful observation? If either of these are the case, why not just game someone else? Why does this particular girl matter? There are billions of them, quite a many of them attractive. If you feel this one isn't worth a perceived risk to approach, drop it and look for another one.
That being said, the point of you posting this post WAS to meet this particular girl, and I'm certainly not here to dissuade you from pursuing her, so here is my advice, broken down as much as possible, and assuming you are a complete stranger to this girl or she only vaguely knows you:
Walking up to a girl you like in front of her friends and striking up a conversation with her with the intention of getting a date with her is difficult, because pulling it off correctly requires very tight fundamentals when in a highly socially charged classroom setting, but it is very possible. However, I would wait for a situation where her friends were not around to talk to her. This is not because it is impossible for you to get her to want to go somewhere with you. This is because at this point if you want to approach her in the actual class, you will either have to:
1. Sit where she usually sits and hope she sits next to you,
2. Sit next to her when she gets there, and open her before her friends get there and say something to make it weird,
*Note, sitting next to her isn't weird in and of itself, but to your credit if the friends get there before you open her, and say or do something to make it SEEM weird for her, she will be MUCH less receptive to you. For example, maybe a dude who usually sits next to her, actually knows her and could like her/be trying to game her as well, sees you in his usual seat and gives her a "what the fuck is this guy doing in my seat, or next to you for that matter?" look when he arrives, which sets a creepy frame for you with ONE look, effectively dealing a huge blow to your chances with her if you haven't opened her yet. Trust me, if he knows her well enough and you don't talk to her before he gets there, this will work for him. This is dirty gaming warfare, (or "strategic cock-blocking", if you will) something I have used before to divert attempts to game a girl I had my eyes on, but am not particularly proud of. Basically, since you cannot rely on her friends to help you ON SIGHT (unless they are all girls, and your social proof at the college is high enough for them to be like "Oh shit, *enter name here* is sitting next to *girl you like*, lets sit somewhere else", you are simply mind-blowingly attractive on sight without speaking to them, or they already know you and know you are attractive), it is much better for you to open her before they show up.
Or lastly,
3. Simply walk up to her with all her friends around before class, which will inevitably involve you openly stating your intentions (think something along the lines of "Hey, I couldn't help but notice how attractive you are, would you want to grab something to eat/grab a coffee/hang out soon/etc"). This just lays everything out on the table, and while this does work sometimes, pulling this off usually requires very tight fundamentals, or high social proof/mind-blowing physical attractiveness, simply because she is forced to make a split second decision about you in front of her friends who she knows will be judging her. If you fail to impress, and she says yes (which she won't unless she is a god damn saint, and there are those girls out there God bless 'um but they are way too few and far between), she has to explain herself to her friends right then. If she says no, her friends will most likely feel bad for you, but congratulate her for dodging the bullet. All of this can be overcome with work, but I would be very confident in your abilities before attempting it this way.
Number 1 is unlikely to lead to positive results because it relies too much on luck.
Number 2 could work, but it would require some luck and precise timing, as well as good opening/conversational skills, and carries the risk of the friends being overbearing enough to actually ask you to move/seeing that the interaction isn't going well and it seems like the girl needs "rescuing", leading to the same result.
Number 3 requires an assload of balls (excuse my unconventional figures of speech) or lots of social proof/physical attractiveness.
Therefore, I recommend that you either do number three in the context of number 2 (state your intentions and ask her out before her friends show up, nail down a date, then get out of dodge) OR "conveniently" run into her somewhere else on campus when she is without company (dining hall lines/library/even walking across campus) and strike up a conversation/ask her out there.
Hope this helps,
Jay