# General semilinear group:GammaL(1,8)

(Redirected from SmallGroup(21,1))
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## Definition

This group is defined in the following equivalent ways:

1. It is the general semilinear group of degree one over field:F8.
2. It is the external semidirect product of cyclic group:Z7 by cyclic group:Z3 for the unique nontrivial action of the latter on the former.

## Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 21 groups with same order As $\Gamma L (1,q), q = p^r, q = 8, p = 2, r = 3$: $r(q - 1) = 3(8 - 1) = 3(7) = 21$
As semidirect product of groups of order 7 and 3: $7 \times 3 = 21$ (see order of semidirect product is product of orders)
exponent of a group 21 groups with same order and exponent of a group | groups with same exponent of a group There are elements of order 7 and 3
nilpotency class -- -- not a nilpotent group
derived length 2 groups with same order and derived length | groups with same derived length Derived subgroup is isomorphic to cyclic group:Z7.
Fitting length 2 groups with same order and Fitting length | groups with same Fitting length Fitting subgroup is same as derived subgroup
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length Maximal subgroups of orders 7 and 3 intersect trivially

### Arithmetic functions of a counting nature

Function Value Similar groups Explanation
number of conjugacy classes 5 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As $\Gamma L (1, p^3), p = 2$: $(p^3 + 8p - 9)/3 = (2^3 + 8 \cdot 2 - 9)/3 = 5$
See element structure of general semilinear group of degree one over a finite field

## GAP implementation

### Group ID

This finite group has order 21 and has ID 1 among the groups of order 21 in GAP's SmallGroup library. For context, there are groups of order 21. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(21,1)

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

gap> G := SmallGroup(21,1);

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

IdGroup(G) = [21,1]

or just do:

IdGroup(G)

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