Holomorph of Q8
Definition
This group can be defined as:
- It is the holomorph of the quaternion group. In other words, it is the semidirect product of the quaternion group with its automorphism group.
Arithmetic functions
Basic arithmetic functions
| Function | Value | Similar groups | Explanation |
|---|---|---|---|
| order (number of elements, equivalently, cardinality or size of underlying set) | 192 | groups with same order | |
| exponent of a group | 24 | groups with same order and exponent of a group | groups with same exponent of a group |
Group properties
Important properties
| Property | Satisfied | Explanation | Comment |
|---|---|---|---|
| abelian group | No | quaternion group is non-abelian, this is a semidirect product with one of the parts involving this non-abelian group |
GAP implementation
Group ID
This finite group has order 192 and has ID 1494 among the groups of order 192 in GAP's SmallGroup library. For context, there are groups of order 192. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(192,1494)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(192,1494);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [192,1494]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Alternative descriptions
It can also be constructed using a hand-coded GAP function: Holomorph, with which it becomes:
Holomorph(SmallGroup(8,4))
as SmallGroup(8,4) is the quaternion group.