# Free product of class two of two Klein four-groups

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## Definition

This group is defined in the following equivalent ways:

1. It is the quotient of the free product of two Klein four-groups by the third member of its lower central series. In other words, it is the free product of two Klein four-groups within the variety of groups of nilpotency class at most two.
2. It is the maximal unipotent subgroup of symplectic group of degree four over field:F4. Equivalently, it is the 2-Sylow subgroup of symplectic group:Sp(4,4).

## GAP implementation

### Group ID

This finite group has order 256 and has ID 8935 among the groups of order 256 in GAP's SmallGroup library. For context, there are groups of order 256. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(256,8935)

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

gap> G := SmallGroup(256,8935);

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

IdGroup(G) = [256,8935]

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
SylowSubgroup(Sp(4,4),2) SylowSubgroup, Sp

### Other descriptions

The group can be constructed as follows:

```gap> F := FreeProduct(ElementaryAbelianGroup(4),ElementaryAbelianGroup(4));
<fp group on the generators [ f1, f2, f3, f4 ]>
gap> G := F/CommutatorSubgroup(F,DerivedSubgroup(F));
Group([ f1, f2, f3, f4 ])```