Holomorph of a group

Symbol-free definition
The holomorph of a group is defined as its external semidirect product with its automorphism group.

Particular cases
Here are some examples for groups of small orders.

Note that for any complete group, i.e., a centerless group in which every automorphism is inner, the holomorph is isomorphic to the square of the group, i.e., its external direct product with itself. This is true even though the action given by the external semidirect product is nontrivial -- what's happening is that if we view the direct product as a semidirect product taking the diagonal subgroup as the complement, we get the holomorph. The smallest nontrivial example group is symmetric group:S3.