General affine group:GA(1,5)
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This group is defined in the following equivalent ways:
- It is the general affine group of degree one over the field of five elements. In other words, it is the semidirect product of the additive and multiplicative groups of this field. It is denoted .
- It is the holomorph of the cyclic group of order five.
- It is the Suzuki group or the Suzuki group where . Note: This is the only non-simple Suzuki group.
The group can be given by the presentation, with denoting the identity element:
This finite group has order 20 and has ID 3 among the groups of order 20 in GAP's SmallGroup library. For context, there are 5 groups of order 20. It can thus be defined using GAP's SmallGroup function as:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(20,3);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [20,3]
or just do:
to have GAP output the group ID, that we can then compare to what we want.