The structure of the gauge group constrains the properties of operators in a G-gauge theory. In this talk, I will consider the reverse direction and ask what properties of a (finite) gauge group can be reconstructed from the operators. I will first consider OPE/fusion rules of Wilson lines and explain various properties of the gauge group that can be reconstructed from it. Then I will introduce certain surface operators which exist in any G-gauge theory. I will describe the fusion rules of these surface operators and show that, in general, there are properties of the gauge group that can be deduced from the fusion rules of surface operators which cannot be obtained from the fusion rules of Wilson lines, and vice-versa.