When you have child,you get something called "allocations familiales",if you have 3 children,you'll get 295€ (460€ for 4,626€ for 5 children and +165€ for every more children.) monthly.
There's also something called "rsa" if you don't work and have children.
You're married with 3 children,so you'll get ~1273€ monthly.
So you'll get ~1568€ monthly. (The minimum wage is 9,53€/hour,if you work 35hours/week,it'll be 1445€ monthly..)
Also,you can get some help by the state monthly to pay your housing,but it depends.)
Also,the free care (reimbursed at 100% by the state)
Also,if you get another children,you'll receive 923€ again 923€ to the next,again again...
There are also a lot more..... (Aid from schools to your children etc..)