Central product of D8 and Z8

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Definition

This group can be described as the central product of dihedral group:D8 and cyclic group:Z8 over a common cyclic central subgroup of order two.

Position in classifications

Type of classification	Position/number in classification
GAP ID	$(32,38)$, i.e., $38^{th}$ among groups of order 32
Hall-Senior number	17 among groups of order 32
Hall-Senior symbol	$32\Gamma _{2}g$

Arithmetic functions

Function	Value	Explanation
order	32
exponent	16
nilpotency class	2
derived length	2
Frattini length	3

Group properties

Function	Value	Explanation
cyclic group	No
abelian group	No
group of nilpotency class two	Yes
metabelian group	Yes

GAP implementation

Group ID

This finite group has order 32 and has ID 38 among the groups of order 32 in GAP's SmallGroup library. For context, there are groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,38)

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

gap> G := SmallGroup(32,38);

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

IdGroup(G) = [32,38]

or just do:

IdGroup(G)

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

Description by presentation

Here is the description by a presentation:

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^8,F.2^2,F.3^2,F.1*F.2*F.1^(-1)*F.2^(-1)];
<fp group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^8,F.2^2,F.3^2,F.1*F.2*F.1^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.1^(-5),F.3*F.2*F.3^(-1)*F.1^4*F.2];
<fp group on the generators [ f1, f2, f3 ]>