Every halachic hour is divided into 1080 chalakim or parts.
e.g. a month is 29 days 12 hours and 793 chalakim. ( I knew that and looked up about the chalakim to remind myself/check about the 1080)
so you could work out 876/1080 for what faction of an hour, then times by 60 to get minutes
52.56 minutes. that's 52 minutes and 33.6 seconds.
so 876 chalakim are 52 minutes 33 seconds, if my maths is right.
I can't comment on the rest of what you wrote re groups of seven.. off the top of my head not sure what that is about(4 days etc). I remember about the difference between the lunar and solar calendar being about 11 days. The lunar year finishing 11 days soon. So an extra lunar month at the end of every third lunar year pushed it out to sync it, the famous 7 lunar months in 19 years(is approx 1 extra lunar month every 3 lunar years).