You might be looking for: black-box group algorithm
A black-box group is essentially a group along with an encoding, along with the caveat that we cannot see inside the algorithms for multiplication, inversion, and validity checking. In other words, these are treated as black boxes.
Black-box group is not a type of group nor is it a particular type of group encoding. Rather, it is a restricted-access view of an encoding of a group.
The notion of black-box group is useful in developing the idea of a black-box group algorithm: an algorithm that works for black-box groups, and hence works for a group with any encoding regardless of what the kind of encoding is.