Combine strings of two columns to create a new column with dplyr R

Question

My data frame looks like this

df <- data.frame(Day=c("Day000","Day000","Day000","Day012","Day012","Day012"), 
                 clones=c("clone01","clone02","clone10","clone11","clone12","clone13"))
df    

     Day  clones
1 Day000 clone01
2 Day000 clone02
3 Day000 clone10
4 Day012 clone11
5 Day012 clone12
6 Day012 clone13

How can I combine the strings of my columns so as my data to look like this

     Day  clones  new.col
1 Day000 clone01   00.01
2 Day000 clone02   00.02
3 Day000 clone10   00.10
4 Day012 clone11   12.11
5 Day012 clone12   12.12
6 Day012 clone13   12.13

Any help or comment are highly appreciated