Moreover, one can discard that without any group-theoretical knowledge: if that was trivial, the other two would also be trivial.

Btw, the task is a bit ill-posed: it is to test our intuition, so we need to think enough to start having an intuition, but need to stop thinking before starting to get the actual answer?

No, no you can think enough to have an intuition and then when you get the actual answer you can test your own intuition. (Thinking about it more, and even working it out is most welcome. Of course, one can look in the literature or wait for the answer.)

Dear Paco, domotorp, and Gabor. You are all correct, of course. Indeed Martin tested us for just two out of the three. (Gabor, Initially I sillingly misunderstood your comment to be: “If its trivial to answer for one group, than it must be trivial to answer for the other groups” 🙂 .)

I believed you are following some pattern trying to show that even though the there is a pattern in relations, there isn’t one in the answers. But maybe I was looking for a pattern where there isn’t one.

One of them is easy to discard, since its abelianization is not trivial. (Gil, feel free to delete this if you think it is too much of a clue)

Moreover, there are two generators and only one equation.

Moreover, one can discard that without any group-theoretical knowledge: if that was trivial, the other two would also be trivial.

Btw, the task is a bit ill-posed: it is to test our intuition, so we need to think enough to start having an intuition, but need to stop thinking before starting to get the actual answer?

No, no you can think enough to have an intuition and then when you get the actual answer you can test your own intuition. (Thinking about it more, and even working it out is most welcome. Of course, one can look in the literature or wait for the answer.)

Dear Paco, domotorp, and Gabor. You are all correct, of course. Indeed Martin tested us for just two out of the three. (Gabor, Initially I sillingly misunderstood your comment to be: “If its trivial to answer for one group, than it must be trivial to answer for the other groups” 🙂 .)

Gil, I think there is a typo. Shouldn’t it be $c^{-1}dc=d^2$ in group C?

GK: yes yes it was a typo. corrected now, thanks YiftachThinking about it, don’t you want also the relation $b^-1ab=a^2$ in A?

Dear Yiftach, what difference does it make?

I believed you are following some pattern trying to show that even though the there is a pattern in relations, there isn’t one in the answers. But maybe I was looking for a pattern where there isn’t one.

Please can someone explain the answer, or tell me where I can find such an explanation?