use of capital C in the word 'Century'

Google ngram shows "twentieth century" - all lower case - is the most common, which is not, of course, the same as saying it is the best or most grammatically correct.

Here are some style recommendations from various more-or-less influential style guides. From The American Heritage Guide to Contemporary Usage and Style (2005):

centuries Centuries may be expressed in words or numerals. [Examples:] words for the twenty-first {or 21st} century; seventeenth-century {or 17th-century} English literature

From The Associated Press Stylebook (2002):

century Lowercase, spelling out numbers less than ten: the first century, the 20th century.

From The Chicago Manual of Style, fifteenth edition (2003):

9.36 Centuries. Particular centuries are spelled out and lowercased. [Examples:] the twenty-first century; the eighth and ninth centuries; the eighteen hundreds (the nineteenth century)

From The Columbia Guide to Standard American English (1993):

century (n.) During this century usually means "since 1900 or 1901" or "within the twentieth century," usually not "within the last hundred years"; a century ago would mean that. And the numbered centuries always encompass the preceding hundred years: the eighteenth century ran from 1700 or 1701 to 1800, the twelfth from 1100 or 1101 to 1200, and so on.

From Bryan Garner, Garner's Modern American Usage, second edition (2003):

CENTURY DESCRIPTIONS. ... The last year of a century ends in 00. But the popular mind has moved everything back a year, in the belief that 2000 marked the beginning of the 21st century. This confusion is unfortunate but seemingly ineradicable.

One other point merits our attention. As compound adjectives, the phrases denoting centuries are hyphenated; but they are not hyphenated as nouns. Hence, "The 12th-century records were discovered in the 19th century."

From MLA Style Manual, second edition (1998):

3.10.5 Dates and Times of the Day ... Spell out centuries in lowercase letters. [Example:] the twentieth century

From The New York Times Manual of Style and Usage, revised edition (1999):

centuries. Lowercase and spell out through the ninth: the eighth century, the 12th century. Hyphenate the adjective form: 18th-century poet. Also, in almost all contexts, the 1700's not seventeen-hundreds; and mid-ninth century, mid-16th century, mid-1890's.

From The Oxford Guide to Style (2002):

Use lower case for millennia and centuries: the first millennium, the sixteenth century. To denote simple ten-year spans OUP style prefers, for example, 1920s or 1960s to nineteen-twenties or nineteen-sixties.

From Andrea Lunsford, The St. Martin's Handbook, fifth edition (2993):

DATES September 17, 1951; the 1860s; the sixties; the '60s; the twentieth century

From Words into Type, third edition (1974):

In referring to centuries, words are preferred to numerals: nineteenth century, twelfth century, twenty-first century.

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Conclusion

Predictably, these ten style guides diverge on a number of points about how to handle references to centuries, decades, and other time periods. But rather astonishingly for a set of competing fashion gurus, all ten agree in preferring to lowercase century in phrases such as "twentieth [or 20th] century"—and I see no reason why they wouldn't also prefer "the present century" (all lowercase).