Faithful semidirect product of E8 and Z4

From Groupprops
Jump to: navigation, search

Definition

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:

\langle x,y,z,a \mid x^2 = y^2 = z^2 = a^4 = e, xy = yx, xz = zx, yz = zy, ax = xa, aya^{-1} = xy, aza^{-1} = yz \rangle

Explicitly, the acting element a is acting as the following matrix if we use \{ x,y,z \} as the basis for elementary abelian group:E8 viewed as a three-dimensional vector space over field:F2:

\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\\end{pmatrix}

Note that this matrix has order four, explaining the group structure.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions
Function Value Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 32 groups with same order
prime-base logarithm of order 5 groups with same prime-base logarithm of order
max-length of a group 5 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 5 chief length equals prime-base logarithm of order for group of prime power order
composition length 5 composition length equals prime-base logarithm of order for group of prime power order
exponent of a group 4 groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent 2 groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class 3 groups with same order and nilpotency class | groups with same nilpotency class
derived length 2 groups with same order and derived length | groups with same derived length
Frattini length 2 groups with same order and Frattini length | groups with same Frattini length
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 3 groups with same order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group 3 groups with same order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group 3 groups with same order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group 3 groups with same order and characteristic rank of a p-group | groups with same characteristic rank of a p-group

Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 32#Group properties
Property Satisfied? Explanation Comment
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
abelian group No

GAP implementation

Group ID

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:

SmallGroup(32,6)

For instance, we can use the following assignment in GAP to create the group and name it G:

gap> G := SmallGroup(32,6);

Conversely, to check whether a given group G is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [32,6]

or just do:

IdGroup(G)

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