This is a question requiring the good knowledge of group theory:
(Q1) Which finite groups $G$ contains some specific centralizers both of these two groups:
i. the elementary group $Z_2^4$, and
ii. the $D_4 \times Z_2$
-
(Q2) Which finite groups $G$ ONLY contains specific centralizers isomorphic to these two groups (but contain NOTHING else):
i. the elementary group $Z_2^4$, and
ii. the $D_4 \times Z_2$
where $D_4$ is dihedral group with the order of $|D_4|=8$. Here $Z_2$ is the cyclic group of the order $|Z_2|=2$.
Let us consider the $|G|$ be as small as possible. Your answer only needs to provide just AN example, the order of the group |G| and its all list of centralizers. (NO need to be complete.) :o)
see also this.