Semidirect product of cyclic group of prime-cube order and cyclic group of prime order
This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups
Definition
This group is defined as the external semidirect product of the cyclic group of order with the unique subgroup of order
in its automorphism group. The group has order
and is sometimes denoted
.
Particular cases
Value of prime ![]() |
Value ![]() |
Corresponding group ![]() |
---|---|---|
2 | 16 | M16 |
3 | 81 | M81 |
GAP implementation
This finite group has order the fourth power of the prime, i.e., , and has ID 6 among the groups of order
in GAP's SmallGroup library. For context, there are 15 groups of order
for odd
and 14 groups of order
for
. It can thus be defined using GAP's SmallGroup function as follows, assuming
is specified beforehand:
SmallGroup(p^4,6)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(p^4,6);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [p^4,6]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.