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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 $D_4 \times Z_2$

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(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.

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  • $\begingroup$ see also this question: math.stackexchange.com/questions/759668/… $\endgroup$ – annie heart Apr 18 '14 at 21:27
  • $\begingroup$ For Q2: $D_4$ contains centralizers $D_4$, $Z_2\times Z_2$ and $Z_4$. If so, we are allowed to include the centralizers of $D_4 \times Z_2$ as $D_4 \times Z_2$, $Z_2 \times Z_2 \times Z_2$ and $Z_4 \times Z_2$. $\endgroup$ – annie heart Apr 18 '14 at 21:35
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    $\begingroup$ SmallGroup(64,138) satisfies (Q1). $\endgroup$ – Derek Holt Apr 18 '14 at 21:54
  • $\begingroup$ Thanks Derek, this is interesting. I have troubled to access GAP so I can only learn that: SmallGroup (64,138) = (((C4 x C2) : C2) : C2) : C2; but what is the form of this group? Can it be further decomposed by a direct product or semi-direct product of elementary groups, cyclic groups or Dihedral groups etc? $\endgroup$ – annie heart Apr 18 '14 at 22:14
  • $\begingroup$ And what is the order of SmallGroup (64,138)? (the number of group elements?) $\endgroup$ – annie heart Apr 18 '14 at 22:24

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