What is the abbreviation for 'century'?

You're asking a question about style, as such you're more likely to find a satisfactory answer outside of the dictionary.

The Associated Press Style Manual doesn't address century specifically, but says you should "Never use an abbreviation that will not be easily understood."

Wikipedia's Manual of Style has a variety of related statements:

• "Centuries and millennia not in quotes or titles should be either spelled out (eighth century) or in Arabic numeral(s) (8th century). The same style should be used throughout any article."
• "To indicate around, approximately, or about, the unitalicised abbreviation c. is preferred over circa, ca, ca., approximately, or approx., and should be spaced (c. 1291). Do not use a question mark for this function (1291?), as this may imply to the reader an uncertainty on the part of Wikipedia editors rather than on the part of reliable historians."
• "Standard symbols for units are undotted; e.g. m for the metre (not m.), kg for the kilogram (not kg.), in for the inch (not in., the quotation mark ", or the double prime ″), and ft for foot (not ft., the apostrophe ', or the prime ′).
  • Non-standard abbreviations should be dotted."

The Association of American Colleges and Universities Style Guide discusses this issue directly: "Centuries are spelled out (CMS 9.33): the twenty-first century"

I'm sure you can find others, but to my way of thinking, your best bet is to either not abbreviate century, or use something unambiguous (such as "cent.") if you have to use an abbreviation.

The Chicago Manual of Style (16th ed., section 10.43) uses c. to abbreviate century, and advises that it is used only "in bibliographic references, glossaries, and other scholarly apparatus." I use in in scholarly book/journal indexes. For example, one sub-entry is "interpretations of (15c.–18c.)"

FWIW, when I was taught (by History teachers) to take lecture notes as a VIth Form A-level student in Britain in the 1970s, we were taught the abbreviation "C12th" etc. as a standard usage (optionally with the C larger than and partially enclosing the number). Several teachers with different academic backgrounds used this form of abbreviation on blackboards, handouts etc. However, we weren't allowed to use it in essays/formal writing. It's a note-taking or informal usage shorthand. I never saw the "15C/c" form until recently on the Internet and I think that is unclear and potentially confusing whereas the capital C plus number with abbreviated ordinal indicator is pretty unequivocal and obvious. I've used the "C20th" form often in varied contexts ever since and can't recall ever being asked to explain it, so it clearly it has good recognition/obvious meaning.

I was taught XXI or 21c. for twenty first century. The most important part being that the reader understands the writers intention.