General semilinear group:GammaL(1,8)
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This group is defined in the following equivalent ways:
- It is the general semilinear group of degree one over field:F8.
- 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.
Basic arithmetic functions
|order (number of elements, equivalently, cardinality or size of underlying set)||21||groups with same order|| As : |
As semidirect product of groups of order 7 and 3: (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
|number of conjugacy classes||5||groups with same order and number of conjugacy classes | groups with same number of conjugacy classes|| As : |
See element structure of general semilinear group of degree one over a finite field
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:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(21,1);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [21,1]
or just do:
to have GAP output the group ID, that we can then compare to what we want.