Semiabelian groups are a class of groups first introduced by Thompson (1984) and named by Matzat (1987).
[1] It appears in Galois theory, in the study of the inverse Galois problem or the embedding problem which is a generalization of the former.
Definition:[2][3][4][5] A finite group G is called semiabelian if and only if there exists a sequence
is a homomorphic image of a semidirect product
of finite semiabelian groups is the minimal family which contains the trivial group and is closed under the following operations:[6][7] The class of finite groups G with a regular realizations over
is the smallest class of finite groups that have both of these closure properties as mentioned above.