Faithful semidirect product of E8 and Z4
This group can be defined as the unique (up to isomorphism) group obtained as the semidirect product of elementary abelian group:E8 and cyclic group:Z4 acting on the elementary abelian group faithfully. An explicit presentation is given by:
Note that this matrix has order four, explaining the group structure.
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions
Want to compare and contrast group properties with other groups of the same order? Check out groups of order 32#Group properties
|group of prime power order||Yes|
|nilpotent group||Yes||prime power order implies nilpotent|
|supersolvable group||Yes||via nilpotent: finite nilpotent implies supersolvable|
|solvable group||Yes||via nilpotent: nilpotent implies solvable|
This finite group has order 32 and has ID 6 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. 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(32,6);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [32,6]
or just do:
to have GAP output the group ID, that we can then compare to what we want.