# SmallGroup(64,17)

View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

This group can be defined by the following presentation: $G := \langle a_1,a_2,a_3 \mid a_1^8 = a_2^4 = a_3^2 = e, [a_1,a_2] = a_3, [a_1,a_3] = [a_2,a_3] = e \rangle$

where $e$ denotes the identity element and $[ , ]$ stands for the commutator of two elements (the isomorphism type of the group is independent of the choice of convention -- left versus right-- for the commutator map, though the specific presentations differ).

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 64#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 64 and has ID 17 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(64,17)

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

gap> G := SmallGroup(64,17);

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

IdGroup(G) = [64,17]

or just do:

IdGroup(G)

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

### Description by presentation

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^8,F.2^4,F.3^2,Comm(F.1,F.2)*F.3^(-1),Comm(F.1,F.3),Comm(F.2,F.3)];
<fp group on the generators [ f1, f2, f3 ]>
gap> IdGroup(G);
[ 64, 17 ]