half-dozen vs six

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

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

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

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

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

half-dozen (noun)	six (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

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