# Inner holomorph of D8

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## Definition

This group can be defined in the following equivalent ways:

1. It is the inner holomorph of the dihedral group of order eight. In other words, it is the semidirect product of the dihedral group by its inner automorphism group, which is isomorphic to a Klein four-group.
2. It is the central product of the dihedral group of order eight with itself, with the common center identified.
3. it is the inner holomorph of the quaternion group. In other words, it is the semidirect product of the quaternion group by its inner automorphism group, which is isomorphic to a Klein four-group.
4. It is the central product of the quaternion group of order eight with itself, with the common center identified.
5. It is the extraspecial group of order $2^5$ and '+' type.
6. It is the subgroup of upper-triangular unipotent matrix group:U(4,2) given by the matrices with only corner entries, i.e., matrices over field:F2 of the form: $\begin{pmatrix} 1 & * & * & * \\ 0 & 1 & 0 & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \\\end{pmatrix}$

The group can also be given by the presentation: $\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz = zx, yz = zy, ax = xa, bx = xb, aya^{-1} = xy, az = za, by = yb, bzb^{-1} = yz \rangle$

## Group properties

Property Satisfied Explanation Comment
group of prime power order Yes
nilpotent group Yes prime power order implies nilpotent
supersolvable group Yes via nilpotent: finite nilpotent implies supersolvable
solvable group Yes via nilpotent: nilpotent implies solvable
abelian group No
extraspecial group Yes
Frattini-in-center group Yes
directly indecomposable group Yes
centrally indecomposable group No
splitting-simple group No

## GAP implementation

### Group ID

This finite group has order 32 and has ID 49 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,49)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(32,49);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [32,49]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Short descriptions

Description Functions used Mathematical comment
ExtraspecialGroup(2^5,'+') ExtraspecialGroup extraspecial group of order $2^5$ and type '+'.

### Long descriptions

Based on the inner holomorph idea, the group can be described as follows using GAP's DihedralGroup, AutomorphismGroup, InnerAutomorphismsAutomorphismGroup and SemidirectProduct functions:

gap> H := DihedralGroup(8);;
gap> A := AutomorphismGroup(H);;
gap> I := InnerAutomorphismsAutomorphismGroup(A);;
gap> G := SemidirectProduct(I,H);
<pc group with 5 generators>