General affine group:GA(1,7)

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Definition

The group is defined in the following equivalent ways:

1. It is the holomorph of the cyclic group of order seven.
2. it is the general affine group of degree one over the field of seven elements.

Properties

It is a semidirect product $(C_{7}\rtimes C_{3})\rtimes C_{2}$. It is a Frobenius group. It is a group of order 42.

Canonical matrix representation of elements

While any general affine group $GA(n,K)$ cannot be realized as a subgroup of the general linear group $GL(n,K)$, it can be realized as a subgroup of $GL(n+1,K)$ in a fairly typical way: the vector from $K^{n}$ is the first $n$ entries of the right column, the matrix from $GL(n,K)$ is the top left $n\times n$ block, there is a $1$ in the bottom right corner, and zeroes elsewhere on the bottom row. In particular, $GA(1,7)$ is the set of matrices over $\mathbb {F} _{7}$ of the form ${\begin{pmatrix}a&b\\0&1\end{pmatrix}}$ with $a\neq 0$.

Arithmetic functions

Function	Value	Explanation
order	42
exponent	42
Frattini length	1
Fitting length	2
derived length	2
subgroup rank	2
minimum size of generating set	2

Group properties

Property	Satisfied	Explanation
abelian group	No
nilpotent group	No
solvable group	Yes
metacyclic group	Yes
supersolvable group	Yes

GAP implementation

Group ID

This finite group has order 42 and has ID 1 among the groups of order 42 in GAP's SmallGroup library. For context, there are groups of order 42. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(42,1)

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

gap> G := SmallGroup(42,1);

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

IdGroup(G) = [42,1]

or just do:

IdGroup(G)

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