Special affine group:SA(2,3)

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Definition

This group is defined as the special affine group of degree two over field:F3.

Name

This group is sometimes called the Hessian group.

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 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>