General affine group:GA(1,7)

From Groupprops
Jump to: navigation, search
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]


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.

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:


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:


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