# Wreath product of D8 and Z2

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## Definition

This group is defined in the following equivalent ways:

1. It is the wreath product of the dihedral group of order eight and the cyclic group of order two with a regular group action.
2. It is the $2$-Sylow subgroup of the symmetric group of degree eight.

## GAP implementation

### Group ID

This finite group has order 128 and has ID 928 among the groups of order 128 in GAP's SmallGroup library. For context, there are groups of order 128. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(128,928)

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

gap> G := SmallGroup(128,928);

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

IdGroup(G) = [128,928]

or just do:

IdGroup(G)

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

### Other descriptions

Description Functions used Mathematical comment
WreathProduct(DihedralGroup(8),CyclicGroup(2)) WreathProduct, DihedralGroup, CyclicGroup
SylowSubgroup(SymmetricGroup(8),2) SylowSubgroup, SymmetricGroup