SmallGroup(48,50)

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
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Definition

This group is a semidirect product $(Z_{2})^{4}\rtimes Z_{3}$. It is given by:

$\langle a,b,c,d,x\mid a^{2}=b^{2}=c^{2}=d^{2}=x^{3}=e,ab=ba,ac=ca,ad=da,bc=cb,bd=db,cd=dc,xax^{-1}=d,xbx^{-1}=bc,xcx^{-1}=b,xdx^{-1}=ad\rangle$

where $e$ is the identity element.

The element $x$ acts as the following matrix if we use $\{a,b,c,d\}$ as the basis for elementary abelian group:E16 viewed as a four-dimensional vector space over field:F2:

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

Note that this matrix has order three, 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 48#Arithmetic functions

Function	Value	Similar groups
order (number of elements, equivalently, cardinality or size of underlying set)	48	groups with same order
exponent of a group	6	groups with same order and exponent of a group | groups with same exponent of a group

GAP implementation

Group ID

This finite group has order 48 and has ID 50 among the groups of order 48 in GAP's SmallGroup library. For context, there are groups of order 48. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(48,50)

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

gap> G := SmallGroup(48,50);

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

IdGroup(G) = [48,50]

or just do:

IdGroup(G)

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