General semilinear group:GammaL(1,8)

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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.