Sorry if the question makes no sense, but I've always had a problem "feeling" semidirect products though I understand them.
Unlike direct products, I can easily point it out when I'm working with a group. This happens when some of the elements seem independent of other elements. Like when I'm adding complex numbers, I can see the imaginary part go with the imaginary part, and the real with the real. Or when multiplying, the magnitude multiplies by the magnitude, and the angle adds with the angle.
But I am not able to develope an intuitive feeling to know when a group is a semidirect product of 2 subgroups. The most thing I've went to so far is to see some elements independent but interact in special cases. Like in the dihedral group, where rotations are rotations, and a reflection is a reflection unless there are 2 reflections which might affect a rotation. But other than that, pointing a semidirect product is not as obvious as direct products. Can someone help me quickly notice when a group is a semidirect product of 2 subgroups?