Google contacts app

I have a Samsung Galaxy S6 active Android version 6.01
I just entered a phone number for someone in my gmail contacts on my laptop. Then I went to the text messages app on my phone, opened a new message and typed in the contact's name, expecting the phone number to pop up - it did not. Her phone number does not show up show up on the phone, either, when I try typing in her first name on the key pad.
So it appears that something is not syncing. I have been working on this for an hour and a half trying to figure out why. The phone indicates that syncing of my email accounts HAS occurred. And I manually synced google just in case (although it had just synced).

I noticed that my contacts app on my phone is generic (as is the phone app) - generic or built in samsung - and saw on forums a reference to a Google Contacts app. I looked for Google Contacts app on phone and it is not there anywhere - and I mean anywhere. I searched on Google Play and there is no such animal.

So I am stuck. Can you help?

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Google contacts app