Inner holomorph of D8

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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:

$\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz = zx, yz = zy, ax = xa, bx = xb, aya^{-1} = xy, az = za, by = yb, bzb^{-1} = yz \rangle$

Arithmetic functions

Function	Value	Similar groups	Explanation
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	32	groups with same order
prime-base logarithm of order	5	groups with same prime-base logarithm of order
exponent of a group	4	groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent	2	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class	2	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length	2	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
Frattini length	2	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set	4	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group	4	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group	3	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group	3	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group	1	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group

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>