six vs half-dozen

six is a noun but half-dozen is not a noun.

Word	Noun	Adjective	Verb	Adverb
six	Yes	Yes	No	No
half-dozen	No	Yes	No	No

As adjectives, six and half-dozen are synonyms defined as:

six and half-dozen: denoting a quantity consisting of six items or units

Other synonyms of six include 6, half dozen, vi.

six (noun)	half-dozen (noun)
a playing card or domino or die whose upward face shows six pips
the cardinal number that is the sum of five and one

six (adjective)	half-dozen (adjective)
denoting a quantity consisting of six items or units	denoting a quantity consisting of six items or units