# Sylow subgroup of holomorph of Z27

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## Definition

This group is defined as the $3$-Sylow subgroup of the holomorph of the cyclic group of order 27. Equivalently, it is the semidirect product of the cyclic group of order 27 and the cyclic group of order 9, where the generator of the latter acts on the former by the $4^{th}$ power map.

It is a particular case of a Sylow subgroup of holomorph of cyclic group of prime-cube order.

## GAP implementation

### Group ID

This finite group has order 243 and has ID 22 among the groups of order 243 in GAP's SmallGroup library. For context, there are groups of order 243. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(243,22)

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

gap> G := SmallGroup(243,22);

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

IdGroup(G) = [243,22]

or just do:

IdGroup(G)

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

### Other descriptions

The group can be constructed as the group G using the following commands, that involve CyclicGroup, Automorphism, SylowSubgorup, and SemidirectProduct.

```gap> C := CyclicGroup(27);
<pc group of size 27 with 3 generators>
gap> A := AutomorphismGroup(C);
<group of size 18 with 5 generators>
gap> S := SylowSubgroup(A,3);
<group>
gap> G := SemidirectProduct(S,C);
<pc group with 5 generators>```