(a*b)*c=4a+6b+3c+8ab+4ac+12bc+16abc

a*(b*c)=2a+6b+9c+8ab+12ac+12bc+16abc

these 2 equations will not be equal and they cant be simplified.

if you proceed to try and find inverse and identity element you will also run into trouble.

therefore the set as defined is not a group