> have you ever wondered why there's a "c" there in that suffix instead of the more regular "-ifiation" that would result from directly combining "-ify" and "-ation"?
Well, I wouldn't wonder about a hypothetical -ifiation, because that would have involved a perceived need to derive a verb in -ate from a verb in -fy, and the English -ate derivational suffix producing verbs doesn't really provide any semantics, so there's no point in applying it to something that's already a verb.
But you do raise a good point about -fication in general; I would expect the form to be -faction, as indeed it is in liquefaction and putrefaction. (Or -fection, as it is in perfection and confection.)
This is because the -io(n) suffix is applied to the stem of the fourth principle part of a Latin verb, and the fourth principle part of facio is factum, not *ficatum. We see many words in -ation because many Latin verbs use the regular fourth principle part stem ending of -at-, and when you combine -io(n) onto that you get -atio(n). But we shouldn't see anything in -ication.
I note that etymonline observes that petrification is "etymologically better than the more common petrifaction". This is interesting for two reasons; first, it must be many decades out of date (at least), and second, it states that the form in -fication is to be preferred to -faction. It also (under the entry for petrify) traces this to a Latin verb petrificare, but wiktionary (generally more up to date) mentions no such verb and attributes the form petrification to a formation in French.
The fact that the combining form of facio, facere in Latin is -ficio, -ficere makes me suspicious of etymonline's description of "-ficare" as "the combining form of facere".