Special affine group:SA(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 as the special affine group of degree two over field:F3.

Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 216 groups with same order As SA(2,q), q = 3: q^2(q^3 - q) = 9 \cdot 24 = 216
number of conjugacy classes 10 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As SA(2,q), q = 3: 2q + 4 = 2(3) + 4 = 10

GAP implementation

Group ID

This finite group has order 216 and has ID 153 among the groups of order 216 in GAP's SmallGroup library. For context, there are 177 groups of order 216. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(216,153)

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

gap> G := SmallGroup(216,153);

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

IdGroup(G) = [216,153]

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
SA(2,3) SA (not an in-built function, follow link to get code)

Alternatively, the group can be defined by the following procedure, using the functions ElementaryAbelianGroup, SubgroupsOfIndexTwo, AutomorphismGroup, and SemidirectProduct:

gap> A := ElementaryAbelianGroup(9);;
gap> B := SubgroupsOfIndexTwo(AutomorphismGroup(A))[1];;
gap> G := SemidirectProduct(B,A);
<pc group with 6 generators>