# General affine group:GA(2,3)

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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## Definition

This group is defined in the following equivalent ways:

1. It is the general affine group of degree two over field:F3.
2. It is the holomorph of elementary abelian group:E9.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 432 groups with same order As $GA(2,q), q = 3$: $q^2(q^2 - 1)(q^2 - q) = 9 \cdot 8 \cdot 6 = 432$
number of conjugacy classes 11 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As $GA(2,q), q = 3$: $q^2 + q - 1 = 3^2 + 3 - 1 = 11$

## GAP implementation

### Group ID

This finite group has order 432 and has ID 734 among the groups of order 432 in GAP's SmallGroup library. For context, there are 775 groups of order 432. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(432,734)

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

gap> G := SmallGroup(432,734);

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

IdGroup(G) = [432,734]

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
Holomorph(ElementaryAbelianGroup(9)) Holomorph (not an in-built GAP command), ElementaryAbelianGroup